2.1 Pressure tensor decomposition

By introducing a unit vector in the direction of the local magnetic field $\hat{\boldsymbol{b}}=\boldsymbol{B}/|\boldsymbol{B}|$ and the gyrofrequency $\unicode[STIX]{x1D6FA}=qB_{0}/(mc)$ , the last term in the pressure tensor equation (2.26) can be rewritten as

(2.29) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}\frac{|\boldsymbol{B}|}{B_{0}}[\hat{\boldsymbol{b}}\times \boldsymbol{p}+(\hat{\boldsymbol{b}}\times \boldsymbol{p})^{\text{T}}].\end{eqnarray}$$

At very low frequencies $\unicode[STIX]{x1D714}\ll \unicode[STIX]{x1D6FA}$ (and at very long spatial scales) this term will dominate and must be equal to zero. The simplest possibility that makes this term to be equal to zero is $\boldsymbol{p}^{\text{iso}}=p\unicode[STIX]{x1D644}$ , where $p$ is the scalar pressure and $\unicode[STIX]{x1D644}$ is the unit matrix. In index notation $p_{ij}^{\text{iso}}=p\unicode[STIX]{x1D6FF}_{ij}$ , and we can verify that

(2.30) $$\begin{eqnarray}\displaystyle & \displaystyle (\hat{\boldsymbol{b}}\times \boldsymbol{p}^{\text{iso}})_{ij}=\unicode[STIX]{x1D716}_{ikl}\hat{b}_{k}p_{lj}^{\text{iso}}=\unicode[STIX]{x1D716}_{ikl}\hat{b}_{k}p\unicode[STIX]{x1D6FF}_{lj}=p\unicode[STIX]{x1D716}_{ikj}\hat{b}_{k}=-p\unicode[STIX]{x1D716}_{ijk}\hat{b}_{k}; & \displaystyle\end{eqnarray}$$

(2.31) $$\begin{eqnarray}\displaystyle & \displaystyle [\hat{\boldsymbol{b}}\times \boldsymbol{p}^{\text{iso}}+(\hat{\boldsymbol{b}}\times \boldsymbol{p}^{\text{iso}})^{\text{T}}]_{ij}=-p\unicode[STIX]{x1D716}_{ijk}\hat{b}_{k}-p\unicode[STIX]{x1D716}_{jik}\hat{b}_{k}=-p\hat{b}_{k}(\unicode[STIX]{x1D716}_{ijk}-\unicode[STIX]{x1D716}_{ijk})=0. & \displaystyle\end{eqnarray}$$

There is however a much more general solution that makes the term (2.29) equal to zero, which is

(2.32) $$\begin{eqnarray}\boldsymbol{p}^{\text{g}}=p_{\Vert }\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}+p_{\bot }(\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}),\end{eqnarray}$$

where the ‘g’ stands for gyrotropic. For quick analytic calculations it is sometimes easier to use $\boldsymbol{p}^{\text{g}}=(p_{\Vert }-p_{\bot })\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}+p_{\bot }\unicode[STIX]{x1D644}$ since we have to deal with $\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}$ only once instead of twice. We need to verify that the term (2.29) indeed disappears,

(2.33) $$\begin{eqnarray}\displaystyle & & \displaystyle (\hat{\boldsymbol{b}}\times \boldsymbol{p}^{\text{g}})_{ij}=\unicode[STIX]{x1D716}_{ikl}\hat{b}_{k}p_{lj}^{\text{g}}=\unicode[STIX]{x1D716}_{ikl}\hat{b}_{k}((p_{\Vert }-p_{\bot })\hat{b}_{l}\hat{b}_{j}+p_{\bot }\unicode[STIX]{x1D6FF}_{lj})\nonumber\\ \displaystyle & & \displaystyle =(p_{\Vert }-p_{\bot })\hat{b}_{j}\underbrace{\unicode[STIX]{x1D716}_{ikl}\hat{b}_{k}\hat{b}_{l}}_{=0}+p_{\bot }\unicode[STIX]{x1D716}_{ikj}\hat{b}_{k}=-p_{\bot }\unicode[STIX]{x1D716}_{ijk}\hat{b}_{k};\end{eqnarray}$$

(2.34) $$\begin{eqnarray}\displaystyle \Rightarrow \quad & & \displaystyle [\hat{\boldsymbol{b}}\times \boldsymbol{p}^{\text{g}}+(\hat{\boldsymbol{b}}\times \boldsymbol{p}^{\text{g}})^{\text{T}}]_{ij}=-p_{\bot }\hat{b}_{k}(\unicode[STIX]{x1D716}_{ijk}+\unicode[STIX]{x1D716}_{jik})=0.\end{eqnarray}$$

It is easy to see that the parallel and perpendicular pressures can be extracted from the gyrotropic pressure tensor matrix by performing double contractions according to

(2.35) $$\begin{eqnarray}p_{\Vert }=\boldsymbol{p}^{\text{g}}\boldsymbol{ : }\hat{\boldsymbol{b}}\hat{\boldsymbol{b}},\quad \text{and}\quad p_{\bot }=\boldsymbol{p}^{\text{g}}\boldsymbol{ : }(\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})/2.\end{eqnarray}$$

The double contraction (the double dot product, represented by the colon) is a very useful operator that is frequently encountered in higher-order fluid models. For two matrices it is defined as $\unicode[STIX]{x1D63C}\boldsymbol{ : }\unicode[STIX]{x1D63D}=\unicode[STIX]{x1D608}_{ij}\unicode[STIX]{x1D609}_{ij}$ and yields a scalar. Sometimes, the double contraction is defined as $\unicode[STIX]{x1D63C}\boldsymbol{ : }\unicode[STIX]{x1D63D}=\unicode[STIX]{x1D608}_{ij}\unicode[STIX]{x1D609}_{ji}$ , which for symmetric matrices is equivalent to the previous definition. As a reminder, the usual matrix product of two matrices is $(\unicode[STIX]{x1D63C}\boldsymbol{\cdot }\unicode[STIX]{x1D63D})_{ij}=\unicode[STIX]{x1D608}_{ik}\unicode[STIX]{x1D609}_{kj}$ and yields a matrix. It is useful to write down the following identities that involve the double contraction,

(2.36) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{b}}\hat{\boldsymbol{b}}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=1; & \displaystyle\end{eqnarray}$$

(2.37) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D644}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=1; & \displaystyle\end{eqnarray}$$

(2.38) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}):\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=0; & \displaystyle\end{eqnarray}$$

(2.39) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D644}:\unicode[STIX]{x1D644}=3; & \displaystyle\end{eqnarray}$$

(2.40) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}):(\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})/2=1. & \displaystyle\end{eqnarray}$$

Also interestingly, the trace of a matrix $\unicode[STIX]{x1D63C}$ can be expressed through the double contraction with $\unicode[STIX]{x1D644}$ , i.e. $\unicode[STIX]{x1D63C}:\unicode[STIX]{x1D644}=\unicode[STIX]{x1D608}_{ij}\unicode[STIX]{x1D6FF}_{ij}=\unicode[STIX]{x1D608}_{ii}=\text{Tr}(\unicode[STIX]{x1D63C})$ . The pressure decomposition (2.35) can now be verified easily. Even more revealing is to apply the decomposition (2.35) directly to the definition of the entire pressure tensor (2.5), which yields

(2.41) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{p}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=m\int \boldsymbol{c}\boldsymbol{c}f\,\text{d}^{3}v:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=m\int (\boldsymbol{v}\boldsymbol{\cdot }\hat{\boldsymbol{b}}-\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}})^{2}f\,\text{d}^{3}v=m\int (v_{\Vert }-u_{\Vert })^{2}f\,\text{d}^{3}v\equiv p_{\Vert }; & \displaystyle\end{eqnarray}$$

(2.42) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{p}:(\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})/2=\frac{m}{2}\int |\boldsymbol{v}_{\bot }-\boldsymbol{u}_{\bot }|^{2}f\,\text{d}^{3}v\equiv p_{\bot }, & \displaystyle\end{eqnarray}$$

where we have used the fact that the magnitude of the 3-D velocity vector can be decomposed as $|\boldsymbol{v}-\boldsymbol{u}|^{2}=|\boldsymbol{v}_{\bot }-\boldsymbol{u}_{\bot }|^{2}+(v_{\Vert }-u_{\Vert })^{2}$ , or equivalently $|\boldsymbol{c}|^{2}=|\boldsymbol{c}_{\bot }|^{2}+c_{\Vert }^{2}$ .

A distribution function is called isotropic when the direction of the velocity vector $\boldsymbol{v}$ does not matter, and the distribution function depends only on magnitude $|\boldsymbol{v}|$ , as for example the Maxwellian distribution $\text{e}^{-|\boldsymbol{v}|^{2}}$ . A distribution function is called gyrotropic when the direction of the perpendicular velocity vector $\boldsymbol{v}_{\bot }$ does not matter, and the function depends only on $|\boldsymbol{v}_{\bot }|$ , as for example the bi-Maxwellian distribution $\text{e}^{-\unicode[STIX]{x1D6FC}_{\Vert }v_{\Vert }^{2}}\text{e}^{-\unicode[STIX]{x1D6FC}_{\bot }|\boldsymbol{v}_{\bot }|^{2}}$ . In another words, the gyrotropic distribution function is isotropic only in its transverse velocity components. The same vocabulary is used for the fluid moments, and a fluid moment is called gyrotropic when it involves integrals over $|\boldsymbol{c}_{\bot }|^{2i}$ , $i=0,1,2,3\ldots \,$ . For the pressure tensor, there are only two possibilities, represented by the parallel and perpendicular pressures (2.41), (2.42).

The part of the pressure tensor extracted by the decomposition (2.35) is therefore called the gyrotropic pressure $\boldsymbol{p}^{\text{g}}$ , sometimes also called the CGL pressure $\boldsymbol{p}^{\text{CGL}}$ . The gyrotropic approximation is sufficient at very long spatial scales, where the Larmor radius ( $=$  the gyroradius) of particles gyrating around the magnetic field is small, and the non-gyrotropic contributions become negligible. However, at sufficiently small spatial scales comparable to the gyroradius, the non-gyrotropic pressure contributions become significant and we represent these through a tensor $\unicode[STIX]{x1D72B}$ , that is called the finite Larmor radius corrections to the (gyrotropic) pressure tensor, or sometimes non-gyrotropic or gyroviscous stress tensor. The entire pressure tensor is thus decomposed according to

(2.43) $$\begin{eqnarray}\displaystyle \boldsymbol{p}=p_{\Vert }\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}+p_{\bot }(\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})+\unicode[STIX]{x1D72B}, & & \displaystyle\end{eqnarray}$$

and for clarity we write down the decomposition explicitly in matrix notation

(2.44) $$\begin{eqnarray}\displaystyle \boldsymbol{p} & = & \displaystyle p_{\Vert }\left(\begin{array}{@{}ccc@{}}\hat{b}_{x}\hat{b}_{x}, & \hat{b}_{x}\hat{b}_{y}, & \hat{b}_{x}\hat{b}_{z}\\ \hat{b}_{y}\hat{b}_{x}, & \hat{b}_{y}\hat{b}_{y}, & \hat{b}_{y}\hat{b}_{z}\\ \hat{b}_{z}\hat{b}_{x}, & \hat{b}_{z}\hat{b}_{y}, & \hat{b}_{z}\hat{b}_{z}\end{array}\right)+p_{\bot }\left(\begin{array}{@{}ccc@{}}1-\hat{b}_{x}\hat{b}_{x}, & -\hat{b}_{x}\hat{b}_{y}, & -\hat{b}_{x}\hat{b}_{z}\\ -\hat{b}_{y}\hat{b}_{x}, & 1-\hat{b}_{y}\hat{b}_{y}, & -\hat{b}_{y}\hat{b}_{z}\\ -\hat{b}_{z}\hat{b}_{x}, & -\hat{b}_{z}\hat{b}_{y}, & 1-\hat{b}_{z}\hat{b}_{z}\end{array}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\left(\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D6F1}_{xx}, & \unicode[STIX]{x1D6F1}_{xy}, & \unicode[STIX]{x1D6F1}_{xz}\\ \unicode[STIX]{x1D6F1}_{yx}, & \unicode[STIX]{x1D6F1}_{yy}, & \unicode[STIX]{x1D6F1}_{yz}\\ \unicode[STIX]{x1D6F1}_{zx}, & \unicode[STIX]{x1D6F1}_{zy}, & \unicode[STIX]{x1D6F1}_{zz}\end{array}\right).\end{eqnarray}$$

In the momentum equation, the pressure tensor enters through its divergence, which is useful to break down into its components. Since $\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})=\hat{\boldsymbol{b}}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{b}})+(\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\hat{\boldsymbol{b}}$ , and $\unicode[STIX]{x1D735}\boldsymbol{\cdot }(p_{\bot }\unicode[STIX]{x1D644})=\unicode[STIX]{x1D735}p_{\bot }$ , the divergence of the pressure tensor calculates as

(2.45) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{p} & = & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }(p_{\bot }\unicode[STIX]{x1D644})+\unicode[STIX]{x1D735}\boldsymbol{\cdot }[(p_{\Vert }-p_{\bot })\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}]+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D735}p_{\bot }+(p_{\Vert }-p_{\bot })[\hat{\boldsymbol{b}}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{b}})+(\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\hat{\boldsymbol{b}}]+\hat{\boldsymbol{b}}(\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735})(p_{\Vert }-p_{\bot })+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}.\end{eqnarray}$$

Importantly, we have seen in (2.41), (2.42) that the scalar parallel and perpendicular pressures are extracted by applying the double contractions to the entire pressure tensor (and not necessarily only at the gyrotropic part). The decomposition (2.43) can be rewritten as

(2.46) $$\begin{eqnarray}\displaystyle \boldsymbol{p}=(\boldsymbol{p}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}+(\boldsymbol{p}:(\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})/2)(\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})+\unicode[STIX]{x1D72B}. & & \displaystyle\end{eqnarray}$$

Applying the double contraction $:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}$ to (2.46) yields the first requirement for the FLR tensor

(2.47) $$\begin{eqnarray}\unicode[STIX]{x1D72B}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=0.\end{eqnarray}$$

Similarly, applying $:(\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})/2$ to (2.46) yields the second requirement for the FLR tensor,

(2.48) $$\begin{eqnarray}\unicode[STIX]{x1D72B}:(\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})=0.\end{eqnarray}$$

By further using the first requirement, the second requirement can be rewritten as

(2.49) $$\begin{eqnarray}\unicode[STIX]{x1D72B}:\unicode[STIX]{x1D644}=\text{Tr}(\unicode[STIX]{x1D72B})=0.\end{eqnarray}$$

By using the pressure decomposition (2.43) in the pressure tensor equation (2.26), and using the fact that the gyrotropic part of pressure satisfies $\hat{\boldsymbol{b}}\times \boldsymbol{p}^{g}+(\hat{\boldsymbol{b}}\times \boldsymbol{p}^{g})^{\text{T}}=0$ , the pressure tensor equation can be rewritten as

(2.50) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{p}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}\boldsymbol{p}+\boldsymbol{q})+\boldsymbol{p}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}+(\boldsymbol{p}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}+\unicode[STIX]{x1D6FA}\frac{|\boldsymbol{B}|}{B_{0}}[\hat{\boldsymbol{b}}\times \unicode[STIX]{x1D72B}+(\hat{\boldsymbol{b}}\times \unicode[STIX]{x1D72B})^{\text{T}}]=0.\end{eqnarray}$$

We can now derive the time dependent equations for parallel and perpendicular pressures by applying the usual double contractions to this pressure tensor equation.

2.2 Parallel pressure equation

We calculate the double contraction with $\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}$ term by term. We will need the following identities

(2.51) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}):\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=0;\quad \unicode[STIX]{x2202}_{k}(\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}):\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=0. & & \displaystyle\end{eqnarray}$$

The first term calculates as

(2.52) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{p}}{\unicode[STIX]{x2202}t}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}} & = & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(p_{\Vert }\hat{b}_{i}\hat{b}_{j}+p_{\bot }(\unicode[STIX]{x1D6FF}_{ij}-\hat{b}_{i}\hat{b}_{j})+\unicode[STIX]{x1D6F1}_{ij})\hat{b}_{i}\hat{b}_{j}=\frac{\unicode[STIX]{x2202}p_{\Vert }}{\unicode[STIX]{x2202}t}\underbrace{|\hat{\boldsymbol{b}}|^{2}|\hat{\boldsymbol{b}}|^{2}}_{=1}\nonumber\\ \displaystyle & & \displaystyle +\,p_{\Vert }\underbrace{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\hat{b}_{i}\hat{b}_{j})\hat{b}_{i}\hat{b}_{j}}_{=0}+\frac{\unicode[STIX]{x2202}p_{\bot }}{\unicode[STIX]{x2202}t}\underbrace{(\unicode[STIX]{x1D6FF}_{ij}-\hat{b}_{i}\hat{b}_{j})\hat{b}_{i}\hat{b}_{j}}_{=0}\nonumber\\ \displaystyle & & \displaystyle +\,p_{\bot }\underbrace{\hat{b}_{i}\hat{b}_{j}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\unicode[STIX]{x1D6FF}_{ij}-\hat{b}_{i}\hat{b}_{j})}_{=0}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\underbrace{\unicode[STIX]{x1D6F1}_{ij}\hat{b}_{i}\hat{b}_{j}}_{=0})-\unicode[STIX]{x1D6F1}_{ij}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\hat{b}_{i}\hat{b}_{j})=\frac{\unicode[STIX]{x2202}p_{\Vert }}{\unicode[STIX]{x2202}t}-\unicode[STIX]{x1D72B}:\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The second term calculates as

(2.53) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}\boldsymbol{p}):\hat{\boldsymbol{b}}\hat{\boldsymbol{b}} & = & \displaystyle \unicode[STIX]{x2202}_{k}(u_{k}p_{ij})\hat{b}_{i}\hat{b}_{j}=\unicode[STIX]{x2202}_{k}(u_{k}\underbrace{p_{ij}\hat{b}_{i}\hat{b}_{j}}_{=p_{\Vert }})-u_{k}p_{ij}\unicode[STIX]{x2202}_{k}(\hat{b}_{i}\hat{b}_{j})=\unicode[STIX]{x2202}_{k}(u_{k}p_{\Vert })-u_{k}p_{\Vert }\underbrace{\hat{b}_{i}\hat{b}_{j}\unicode[STIX]{x2202}_{k}(\hat{b}_{i}\hat{b}_{j})}_{=0}\nonumber\\ \displaystyle & & \displaystyle -p_{\bot }u_{k}\underbrace{(\unicode[STIX]{x1D6FF}_{ij}-\hat{b}_{i}\hat{b}_{j})\unicode[STIX]{x2202}_{k}(\hat{b}_{i}\hat{b}_{j})}_{=0}-\unicode[STIX]{x1D6F1}_{ij}u_{k}\unicode[STIX]{x2202}_{k}(\hat{b}_{i}\hat{b}_{j})=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(p_{\Vert }\boldsymbol{u})-\unicode[STIX]{x1D72B}:(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}(\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}));\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.54) $$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}):\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q})_{ij}\hat{b}_{i}\hat{b}_{j}=\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q})\boldsymbol{\cdot }\hat{\boldsymbol{b}}. & & \displaystyle\end{eqnarray}$$

The third term calculates as

(2.55) $$\begin{eqnarray}\displaystyle (\boldsymbol{p}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}):\hat{\boldsymbol{b}}\hat{\boldsymbol{b}} & = & \displaystyle (\boldsymbol{p}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})_{ij}\hat{b}_{i}\hat{b}_{j}=(p_{ik}\unicode[STIX]{x2202}_{k}u_{j})\hat{b}_{i}\hat{b}_{j}=(\unicode[STIX]{x2202}_{k}u_{j})p_{ik}\hat{b}_{i}\hat{b}_{j}\nonumber\\ \displaystyle & = & \displaystyle (\unicode[STIX]{x2202}_{k}u_{j})[p_{\Vert }\hat{b}_{i}\hat{b}_{k}+p_{\bot }(\unicode[STIX]{x1D6FF}_{ik}-\hat{b}_{i}\hat{b}_{k})+\unicode[STIX]{x1D6F1}_{ik}]\hat{b}_{i}\hat{b}_{j}\nonumber\\ \displaystyle & = & \displaystyle (\unicode[STIX]{x2202}_{k}u_{j})[p_{\Vert }\hat{b}_{k}\hat{b}_{j}+p_{\bot }\underbrace{(\hat{b}_{k}\hat{b}_{j}-\hat{b}_{k}\hat{b}_{j})}_{=0}+\unicode[STIX]{x1D6F1}_{ik}\hat{b}_{i}\hat{b}_{j}]=p_{\Vert }\hat{b}_{k}(\unicode[STIX]{x2202}_{k}u_{j})\hat{b}_{j}+\unicode[STIX]{x1D6F1}_{ik}(\unicode[STIX]{x2202}_{k}u_{j})\hat{b}_{i}\hat{b}_{j}\nonumber\\ \displaystyle & = & \displaystyle p_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}):\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}.\end{eqnarray}$$

The fourth term calculates similarly as

(2.56) $$\begin{eqnarray}\displaystyle (\boldsymbol{p}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}} & = & \displaystyle (\boldsymbol{p}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})_{ji}\hat{b}_{i}\hat{b}_{j}=(p_{jk}\unicode[STIX]{x2202}_{k}u_{i})\hat{b}_{i}\hat{b}_{j}=(\unicode[STIX]{x2202}_{k}u_{i})p_{jk}\hat{b}_{i}\hat{b}_{j}\nonumber\\ \displaystyle & = & \displaystyle (\unicode[STIX]{x2202}_{k}u_{i})[p_{\Vert }\hat{b}_{j}\hat{b}_{k}+p_{\bot }(\unicode[STIX]{x1D6FF}_{jk}-\hat{b}_{j}\hat{b}_{k})+\unicode[STIX]{x1D6F1}_{jk}]\hat{b}_{i}\hat{b}_{j}\nonumber\\ \displaystyle & = & \displaystyle (\unicode[STIX]{x2202}_{k}u_{i})[p_{\Vert }\hat{b}_{k}\hat{b}_{i}+p_{\bot }\underbrace{(\hat{b}_{i}\hat{b}_{k}-\hat{b}_{k}\hat{b}_{i})}_{=0}+\unicode[STIX]{x1D6F1}_{jk}\hat{b}_{i}\hat{b}_{j}]=p_{\Vert }\hat{b}_{k}(\unicode[STIX]{x2202}_{k}u_{i})\hat{b}_{i}+\unicode[STIX]{x1D6F1}_{jk}(\unicode[STIX]{x2202}_{k}u_{i})\hat{b}_{i}\hat{b}_{j}\nonumber\\ \displaystyle & = & \displaystyle p_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})_{ji}\hat{b}_{i}\hat{b}_{j}=p_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}},\end{eqnarray}$$

and the final fifth term becomes

(2.57) $$\begin{eqnarray}\displaystyle [\hat{\boldsymbol{b}}\times \unicode[STIX]{x1D72B}+(\hat{\boldsymbol{b}}\times \unicode[STIX]{x1D72B})^{\text{T}}]:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}} & = & \displaystyle [(\hat{\boldsymbol{b}}\times \unicode[STIX]{x1D72B})_{ij}+(\hat{\boldsymbol{b}}\times \unicode[STIX]{x1D72B})_{ji}]\hat{b}_{i}\hat{b}_{j}=[\unicode[STIX]{x1D716}_{ikl}\hat{b}_{k}\unicode[STIX]{x1D6F1}_{lj}+\unicode[STIX]{x1D716}_{jkl}\hat{b}_{k}\unicode[STIX]{x1D6F1}_{li}]\hat{b}_{i}\hat{b}_{j}\nonumber\\ \displaystyle & = & \displaystyle \hat{b}_{j}\unicode[STIX]{x1D6F1}_{lj}\underbrace{\unicode[STIX]{x1D716}_{ikl}\hat{b}_{i}\hat{b}_{k}}_{=0}+\hat{b}_{i}\unicode[STIX]{x1D6F1}_{li}\underbrace{\unicode[STIX]{x1D716}_{jkl}\hat{b}_{j}\hat{b}_{k}}_{=0}=0.\end{eqnarray}$$

The entire equation for the parallel pressure therefore reads

(2.58) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}p_{\Vert }}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(p_{\Vert }\boldsymbol{u})+2p_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q})\boldsymbol{\cdot }\hat{\boldsymbol{b}}-\unicode[STIX]{x1D72B}:\left[\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\right)(\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,[\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}+(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}]:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=0.\end{eqnarray}$$

Using the symmetric operator (2.27) and the definition for the convective derivative $\text{d}/\text{d}t\equiv \unicode[STIX]{x2202}/\unicode[STIX]{x2202}t+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ , the equation is written in a simple form as

(2.59) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}p_{\Vert }}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(p_{\Vert }\boldsymbol{u})+2p_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q})\boldsymbol{\cdot }\hat{\boldsymbol{b}}-\unicode[STIX]{x1D72B}:\frac{\text{d}}{\text{d}t}(\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})+(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{S}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=0, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

which corresponds to (26) of Passot & Sulem (Reference Passot and Sulem2004), equation (9) of Goswami et al. (Reference Goswami, Passot and Sulem2005) (see also Oraevskii et al. (Reference Oraevskii, Chodura and Feneberg1968), Passot & Sulem (Reference Passot and Sulem2007) and Passot, Sulem & Hunana (Reference Passot, Sulem and Hunana2012)).

2.3 Perpendicular pressure equation

It is possible to either apply the double contraction  $:(\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})/2$ to the pressure tensor equation to directly obtain the equation for $\unicode[STIX]{x2202}p_{\bot }/\unicode[STIX]{x2202}t$ , or to apply the trace (the double contraction  $:\unicode[STIX]{x1D644}$ ) to obtain the equation for $\unicode[STIX]{x2202}p_{\Vert }/\unicode[STIX]{x2202}t+2\unicode[STIX]{x2202}p_{\bot }/\unicode[STIX]{x2202}t$ , and then subtract the previously obtained equation for $\unicode[STIX]{x2202}p_{\Vert }/\unicode[STIX]{x2202}t$ . Both approaches are equivalent, since $p_{\bot }=(\text{Tr}(\boldsymbol{p})-p_{\Vert })/2$ . In another words, directly calculating the trace of the pressure tensor yields

(2.60) $$\begin{eqnarray}\text{Tr}\,\boldsymbol{p}=\unicode[STIX]{x1D6FF}_{ij}p_{ij}=p_{ii}=p_{\Vert }\underbrace{\hat{b}_{i}\hat{b}_{i}}_{=1}+p_{\bot }\underbrace{(\unicode[STIX]{x1D6FF}_{ii}-\hat{b}_{i}\hat{b}_{i})}_{=2}+\underbrace{\unicode[STIX]{x1D6F1}_{ii}}_{=0}=p_{\Vert }+2p_{\bot },\end{eqnarray}$$

where we have used that $\text{Tr}\,\unicode[STIX]{x1D72B}=\unicode[STIX]{x1D6F1}_{ii}=0$ . Applying the trace operator to the pressure tensor equation (2.50) term by term yields for the first term

(2.61) $$\begin{eqnarray}\displaystyle \text{Tr}\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\boldsymbol{p}=\unicode[STIX]{x1D6FF}_{ij}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}p_{ij}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}p_{ii}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(p_{\Vert }+2p_{\bot }); & & \displaystyle\end{eqnarray}$$

for the second term

(2.62) $$\begin{eqnarray}\displaystyle \text{Tr}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{u}\boldsymbol{p})) & = & \displaystyle \unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x2202}_{k}(u_{k}p_{ij})=\unicode[STIX]{x2202}_{k}(u_{k}p_{ii})=u_{k}\unicode[STIX]{x2202}_{k}p_{ii}+p_{ii}\unicode[STIX]{x2202}_{k}u_{k}\nonumber\\ \displaystyle & = & \displaystyle \boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}(p_{\Vert }+2p_{\bot })+(p_{\Vert }+2p_{\bot })\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u};\end{eqnarray}$$

(2.63) $$\begin{eqnarray}\displaystyle \text{Tr}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q})=\text{unchanged}; & & \displaystyle\end{eqnarray}$$

the third term

(2.64) $$\begin{eqnarray}\displaystyle \text{Tr}(\boldsymbol{p}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}) & = & \displaystyle \unicode[STIX]{x1D6FF}_{ij}(\boldsymbol{p}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})_{ij}=p_{ik}\unicode[STIX]{x2202}_{k}u_{i}=((p_{\Vert }-p_{\bot })\hat{b}_{i}\hat{b}_{k}+p_{\bot }\unicode[STIX]{x1D6FF}_{ik}+\unicode[STIX]{x1D6F1}_{ik})\unicode[STIX]{x2202}_{k}u_{i}\nonumber\\ \displaystyle & = & \displaystyle (p_{\Vert }-p_{\bot })\hat{b}_{k}(\unicode[STIX]{x2202}_{k}u_{i})\hat{b}_{i}+p_{\bot }\unicode[STIX]{x2202}_{i}u_{i}+\unicode[STIX]{x1D6F1}_{ik}\unicode[STIX]{x2202}_{k}u_{i}\nonumber\\ \displaystyle & = & \displaystyle (p_{\Vert }-p_{\bot })\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+p_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}+\text{Tr}(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u});\end{eqnarray}$$

the fourth term is equivalent to the third term

(2.65) $$\begin{eqnarray}\displaystyle \text{Tr}(\boldsymbol{p}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}=(\boldsymbol{p}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})_{ii}=(p_{\Vert }-p_{\bot })\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+p_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}+\text{Tr}(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}; & & \displaystyle\end{eqnarray}$$

even though we kept the last expression in transpose form so that we can use the symmetric operator after we combine all the terms. Finally the fifth term

(2.66) $$\begin{eqnarray}\displaystyle & \displaystyle \text{Tr}(\hat{\boldsymbol{b}}\times \unicode[STIX]{x1D72B})=\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D716}_{ikl}\hat{b}_{k}\unicode[STIX]{x1D6F1}_{lj}=\hat{b}_{k}\underbrace{\unicode[STIX]{x1D716}_{ikl}\unicode[STIX]{x1D6F1}_{li}}_{=0}=0; & \displaystyle\end{eqnarray}$$

(2.67) $$\begin{eqnarray}\displaystyle & \displaystyle \text{Tr}(\hat{\boldsymbol{b}}\times \unicode[STIX]{x1D72B})^{\text{T}}=\text{Tr}(\hat{\boldsymbol{b}}\times \unicode[STIX]{x1D72B})=0. & \displaystyle\end{eqnarray}$$

Combining all the terms, using the parallel pressure equation (2.59) and dividing by 2 yields the time-dependent equation for the perpendicular pressure that reads

(2.68) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}p_{\bot }}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(p_{\bot }\boldsymbol{u})+p_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}-p_{\bot }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+\frac{1}{2}[\text{Tr}\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}-\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q})\boldsymbol{\cdot }\hat{\boldsymbol{b}}]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{1}{2}\left[\text{Tr}(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{S}+\unicode[STIX]{x1D72B}:\frac{\text{d}}{\text{d}t}(\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})-(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{S}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}\right]=0.\end{eqnarray}$$

The result corresponds to, for example, equation (25) of Passot & Sulem (Reference Passot and Sulem2004), equation (8) of Goswami et al. (Reference Goswami, Passot and Sulem2005). It is important to note that the parallel and perpendicular pressure equations are exact equations. These equations are valid regardless of what kind of higher-order closure will be adopted later. It is useful to rewrite the term $\unicode[STIX]{x1D72B}:(\text{d}/\text{d}t)(\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})$ slightly so that we can directly use the induction equation and eliminate the time derivative. The term can be rewritten as

(2.69) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D72B}:\frac{\text{d}}{\text{d}t}(\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}) & = & \displaystyle \unicode[STIX]{x1D6F1}_{ij}\frac{\text{d}}{\text{d}t}(\hat{b}_{i}\hat{b}_{j})=\unicode[STIX]{x1D6F1}_{ij}\left(\hat{b}_{i}\frac{\text{d}}{\text{d}t}\hat{b}_{j}+\hat{b}_{j}\frac{\text{d}}{\text{d}t}\hat{b}_{i}\right)\nonumber\\ \displaystyle & = & \displaystyle \hat{b}_{i}\unicode[STIX]{x1D6F1}_{ij}\frac{\text{d}}{\text{d}t}\hat{b}_{j}+\hat{b}_{j}\underbrace{\unicode[STIX]{x1D6F1}_{ij}}_{\unicode[STIX]{x1D6F1}_{ji}}\frac{\text{d}}{\text{d}t}\hat{b}_{i}=2\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\frac{\text{d}\hat{\boldsymbol{b}}}{\text{d}t}.\end{eqnarray}$$

We will need the following identity for the time derivative of the unit vector (see later in the text),

(2.70) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\hat{\boldsymbol{b}}=\frac{1}{|\boldsymbol{B}|}\left[\frac{\text{d}\boldsymbol{B}}{\text{d}t}-\hat{\boldsymbol{b}}\left(\hat{\boldsymbol{b}}\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{B}}{\text{d}t}\right)\right], & & \displaystyle\end{eqnarray}$$

which, when used in the above expression together with $\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\hat{\boldsymbol{b}}=0$ , allows us to write

(2.71) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D72B}:\frac{\text{d}}{\text{d}t}(\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})=\frac{2}{|\boldsymbol{B}|}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{B}}{\text{d}t}, & & \displaystyle\end{eqnarray}$$

which is an expression of general validity since no specific form of the induction equation was assumed yet. In the next section we will calculate the decomposition of the heat flux tensor $\boldsymbol{q}$ and we will show that, if only the gyrotropic heat flux components $q_{\Vert },q_{\bot }$ are considered, the heat flux terms entering the pressure equations read

(2.72) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}^{\text{g}})\boldsymbol{\cdot }\hat{\boldsymbol{b}}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(q_{\Vert }\hat{\boldsymbol{b}})-2q_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{b}}; & \displaystyle\end{eqnarray}$$

(2.73) $$\begin{eqnarray}\displaystyle & \displaystyle {\textstyle \frac{1}{2}}[\text{Tr}\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}^{\text{g}}-\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}^{\text{g}})\boldsymbol{\cdot }\hat{\boldsymbol{b}}]=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(q_{\bot }\hat{\boldsymbol{b}})+q_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{b}}. & \displaystyle\end{eqnarray}$$

Therefore, by splitting the heat flux into gyrotropic and non-gyrotropic parts, $\boldsymbol{q}=\boldsymbol{q}^{\text{g}}+\boldsymbol{q}^{\text{ng}}$ , the pressure equations read

(2.74) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}p_{\Vert }}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(p_{\Vert }\boldsymbol{u})+2p_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(q_{\Vert }\hat{\boldsymbol{b}})-2q_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}^{\text{ng}})\boldsymbol{\cdot }\hat{\boldsymbol{b}}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{2}{|\boldsymbol{B}|}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{B}}{\text{d}t}+(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{S}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=0;\end{eqnarray}$$

(2.75) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}p_{\bot }}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(p_{\bot }\boldsymbol{u})+p_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}-p_{\bot }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(q_{\bot }\hat{\boldsymbol{b}})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,q_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+\frac{1}{2}[\text{Tr}\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}^{\text{ng}}-\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}^{\text{ng}})\boldsymbol{\cdot }\hat{\boldsymbol{b}}]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{1}{2}\left[\text{Tr}(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{S}+\frac{2}{|\boldsymbol{B}|}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{B}}{\text{d}t}-(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{S}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}\right]=0.\end{eqnarray}$$

Terms with the non-gyrotropic heat flux $\boldsymbol{q}^{\text{ng}}$ can be further split into non-gyrotropic heat flux vectors $\boldsymbol{S}_{\bot }^{\Vert }$ , $\boldsymbol{S}_{\bot }^{\bot }$ and the heat flux tensor $\unicode[STIX]{x1D748}$ , which is addressed in appendix D, see (D 124), (D 126). Terms containing the FLR pressure tensor $\unicode[STIX]{x1D72B}$ are usually called the FLR stress forces, and these forces can generate complicated plasma heating processes. The FLR stress forces can generate both parallel and perpendicular plasma heating that can be determined only from fully nonlinear numerical simulations, since all the terms disappear at the linear level. This process is also sometimes called ‘stochastic heating’. The importance of stochastic heating was shown by Laveder, Passot & Sulem (Reference Laveder, Passot and Sulem2013), who performed 1.5-dimensional numerical simulations with the sophisticated FLR Landau fluid model of Passot & Sulem (Reference Passot and Sulem2007), Passot et al. (Reference Passot, Sulem and Hunana2012).

2.4 On the paper by Chew et al. (Reference Chew, Goldberger and Low1956)

The very influential paper by Chew et al. (Reference Chew, Goldberger and Low1956) that is typically cited for the CGL equations (with zero heat fluxes) actually also derived the pressure equations when the gyrotropic heat fluxes are considered. It is important to emphasize that, even though the scalar pressures $p_{n}$ , $p_{s}$ in the notation of that paper are equal to the usual pressures $p_{\Vert }$ , $p_{\bot }$ , this is not the case for the heat fluxes. The gyrotropic heat flux tensor in that paper is decomposed into components $q_{n}$ , $q_{s}$ according to (Chew et al. Reference Chew, Goldberger and Low1956, equation (34))

(2.76) $$\begin{eqnarray}q_{ijk}^{\text{g}}=q_{n}\hat{b}_{i}\hat{b}_{j}\hat{b}_{k}+q_{s}(\unicode[STIX]{x1D6FF}_{ij}\hat{b}_{k}+\unicode[STIX]{x1D6FF}_{jk}\hat{b}_{i}+\unicode[STIX]{x1D6FF}_{ik}\hat{b}_{j}).\end{eqnarray}$$

In contrast, the usual gyrotropic heat flux decomposition to $q_{\Vert }$ , $q_{\bot }$ reads (see later in the text)

(2.77) $$\begin{eqnarray}q_{ijk}^{\text{g}}=q_{\Vert }\hat{b}_{i}\hat{b}_{j}\hat{b}_{k}+q_{\bot }(\unicode[STIX]{x1D6FF}_{ij}\hat{b}_{k}+\unicode[STIX]{x1D6FF}_{jk}\hat{b}_{i}+\unicode[STIX]{x1D6FF}_{ik}\hat{b}_{j}-3\hat{b}_{i}\hat{b}_{j}\hat{b}_{k}),\end{eqnarray}$$

i.e. the last term is not present in their decomposition. However, this does not mean that their pressure equations (Chew et al. Reference Chew, Goldberger and Low1956, equations (31), (32)) are incorrect. It only implies that their equations are written in terms of

(2.78) $$\begin{eqnarray}q_{n}=q_{\Vert }-3q_{\bot };\quad q_{s}=q_{\bot }.\end{eqnarray}$$

By neglecting the FLR corrections $\unicode[STIX]{x1D72B}$ and $\boldsymbol{q}^{\text{ng}}$ , the parallel and perpendicular pressure equations (2.74), (2.75) simplify to

(2.79) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}p_{\Vert }}{\text{d}t}+p_{\Vert }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}+2p_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(q_{\Vert }\hat{\boldsymbol{b}})-2q_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{b}}=0; & \displaystyle\end{eqnarray}$$

(2.80) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}p_{\bot }}{\text{d}t}+2p_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}-p_{\bot }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(q_{\bot }\hat{\boldsymbol{b}})+q_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{b}}=0. & \displaystyle\end{eqnarray}$$

By using (2.78), the heat flux contributions in the parallel pressure equation above can be rewritten as

(2.81) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }(q_{\Vert }\hat{\boldsymbol{b}})-2q_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{b}}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }[(q_{n}+q_{s})\hat{\boldsymbol{b}}]+2\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}q_{s}, & & \displaystyle\end{eqnarray}$$

which agrees with the parallel pressure equation obtained by Chew et al. (Reference Chew, Goldberger and Low1956), equation (31). The pressure equations of Chew et al. (Reference Chew, Goldberger and Low1956) are therefore correct, but it has to be remembered that $q_{n}$ is not the usual parallel heat flux $q_{\Vert }=m\!\int (v_{\Vert }-u_{\Vert })^{3}f\,\text{d}^{3}v$ .Footnote ¹ Actually, the authors do not even call the quantities $q_{n},q_{s}$ heat flux components, but components of a ‘pressure-transport’ tensor.

To conclude this subsection, we will go slightly ahead and use (3.51), (3.52) (without the zero right-hand sides) that are the usual CGL pressure equations and valid only at long spatial scales when the Hall term in the induction equation is neglected. Under this assumption, the pressure equations (2.79), (2.80) are rewritten as

(2.82) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{p_{\Vert }|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{3}}\right)=-\frac{|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{3}}[\unicode[STIX]{x1D735}\boldsymbol{\cdot }(q_{\Vert }\hat{\boldsymbol{b}})-2q_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{b}}]; & \displaystyle\end{eqnarray}$$

(2.83) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{p_{\bot }}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\right)=-\frac{1}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}[\unicode[STIX]{x1D735}\boldsymbol{\cdot }(q_{\bot }\hat{\boldsymbol{b}})+q_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{b}}]. & \displaystyle\end{eqnarray}$$

It is often forgotten that the paper by Chew et al. (Reference Chew, Goldberger and Low1956) derived the correct pressure equations with gyrotropic heat flux contributions. The authors, however, did not write the equations in the form (2.82), (2.83), and left them in the form (2.79), (2.80), using the ‘conservative’ form only after prescribing the closure $q_{\Vert }=q_{\bot }=0$ (their equations (35), (36)),

(2.84) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{p_{\Vert }|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{3}}\right)=0; & \displaystyle\end{eqnarray}$$

(2.85) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{p_{\bot }}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\right)=0. & \displaystyle\end{eqnarray}$$

Over time, equations (2.84), (2.85) became known as the CGL description. Equations (2.84), (2.85) are often replaced by the equations of state $p_{\Vert }|\boldsymbol{B}|^{2}/\unicode[STIX]{x1D70C}^{3}=\text{const.}$ and $p_{\bot }/(\unicode[STIX]{x1D70C}|\boldsymbol{B}|)=\text{const.}$ , see e.g. Mjølhus (Reference Mjølhus2009). Additionally, similarly to the definition of (appropriately normalized) entropy in MHD, $s=\ln (p/\unicode[STIX]{x1D70C}^{\unicode[STIX]{x1D6FE}})$ , it is useful to define parallel and perpendicular entropy in the CGL model, according to

(2.86) $$\begin{eqnarray}s_{\Vert }=\frac{1}{3}\ln \left(\frac{p_{\Vert }|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{3}}\right);\quad s_{\bot }=\frac{2}{3}\ln \left(\frac{p_{\bot }}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\right).\end{eqnarray}$$

The total entropy, $s_{\Vert }+s_{\bot }=\ln (p_{\Vert }^{1/3}p_{\bot }^{2/3}\unicode[STIX]{x1D70C}^{-5/3})$ , is equivalent to the MHD entropy when $p_{\Vert }=p_{\bot }$ , see for example (9)–(11) in Abraham-Shrauner (Reference Abraham-Shrauner1967).

2.5 Physical meaning of CGL equations

The exact derivation of the CGL equation (2.84), (2.85) will be completed in the next section. Here, we briefly discuss the physical meaning of these equations. The CGL equations (2.85), (2.84) can be interpreted as the conservation of the first and second adiabatic invariants, as is nicely discussed for example by Kulsrud (Reference Kulsrud, Galeev and Sudan1983) and by Gurnett & Bhattacharjee (Reference Gurnett and Bhattacharjee2005). Considering many particles with velocity components $v_{\Vert }$ and $v_{\bot }$ with respect to the mean magnetic field, the parallel and perpendicular pressures can be estimated as

(2.87) $$\begin{eqnarray}p_{\Vert }\sim \langle v_{\Vert }^{2}\rangle \unicode[STIX]{x1D70C};\quad p_{\bot }\sim \langle v_{\bot }^{2}\rangle \unicode[STIX]{x1D70C},\end{eqnarray}$$

where the brackets represent an average over all particles. To complete the estimate, we need expressions for $\langle v_{\Vert }^{2}\rangle$ and $\langle v_{\bot }^{2}\rangle$ , which come from the conservation of the adiabatic invariants. The first adiabatic invariant is the conservation of magnetic moment $\unicode[STIX]{x1D707}$ of a particle that is gyrating periodically around a mean magnetic field. The second adiabatic invariant, sometimes also called the longitudinal invariant, is associated with a particle bouncing periodically between two magnetic mirror points. The conserved quantity is the integral over the parallel momentum of a particle $J\equiv \oint _{a}^{b}mv_{\Vert }\,\text{d}l$ , where the magnetic mirrors are located at ‘ $a$ ’ and ‘ $b$ ’ and the integral is performed along the magnetic field line. The distance between the two magnetic mirrors at ‘ $a$ ’ and ‘ $b$ ’ can be labelled as $L$ . The conservation of the first and second adiabatic invariants is therefore

(2.88) $$\begin{eqnarray}\unicode[STIX]{x1D707}\equiv \frac{mv_{\bot }^{2}}{2B}=\text{const.};\quad J\equiv mv_{\Vert }L=\text{const.}\end{eqnarray}$$

Since we are considering only non-relativistic particles, the particle mass ‘ $m$ ’ is also a constant. The conservation of magnetic moment $\unicode[STIX]{x1D707}$ therefore implies $v_{\bot }^{2}\sim B$ , and using this in (2.87) yields $p_{\bot }\sim B\unicode[STIX]{x1D70C}$ , recovering (2.85). The CGL equation (2.85) therefore corresponds to conservation of magnetic moment, i.e. the first adiabatic invariant. The conservation of the second adiabatic invariant $J$ is more tricky, since we need to somehow estimate the non-intuitive length $L$ between the two magnetic mirrors, and no magnetic mirrors are explicitly assumed, since we are just dealing with somewhat random magnetic field lines. The length $L$ can be nevertheless estimated from two fundamental physical principles. Consider a magnetic flux tube (a deformed cylinder) with cross-sectional area $A$ and length $L$ . The conservation of the total magnetic flux through the area $A$ implies $AB=\text{const.}$ , which yields an estimate for the area $A\sim 1/B$ . The second principle is the conservation of the total mass of particles that are completely trapped in that flux tube and that cannot escape, $mnAL=\text{const.}$ , yielding $L\sim 1/(\unicode[STIX]{x1D70C}A)$ . The use of $A\sim 1/B$ yields the final estimate for the non-intuitive length $L\sim B/\unicode[STIX]{x1D70C}$ . The conservation of the second adiabatic invariant $J$ therefore implies $v_{\Vert }\sim 1/L\sim \unicode[STIX]{x1D70C}/B$ . Using this result in (2.87) implies $p_{\Vert }\sim \unicode[STIX]{x1D70C}^{3}/B^{2}$ , recovering (2.84). The CGL equation (2.84) corresponds to conservation of the second adiabatic invariant $J$ , and the equation is valid for particles that are completely trapped and therefore cannot carry a heat flux. As we discuss later, the consideration of the Hall term, the heat flux and the FLR stress forces (the stochastic heating), leads to the breaking of these two adiabatic invariants. Nevertheless, exact conservation laws can still be derived, and the very simple CGL equations (2.84), (2.85) have to be modified with non-zero right-hand sides.

In Part 2 of our guide (see § 4.1), we show that the conservation of magnetic moment is very useful for understanding the form of the perturbed distribution function $f^{(1)}$ in the gyrotropic limit. By performing calculations in the laboratory reference frame, we show that to obtain the correct $f^{(1)}$ in the gyrotropic limit actually requires the usual complicated kinetic integration around the unperturbed orbit (see the Appendix of Part 2) and only then the gyrotropic limit can be imposed. In contrast, performing calculations in the guiding-centre reference frame allows one to prescribe the conservation of magnetic moment from the beginning, and obtain the correct $f^{(1)}$ in the gyrotropic limit perhaps more intuitively. Conservation of adiabatic invariants is important in many areas of space physics and astrophysics. For example, conservation of adiabatic invariants is used to construct models that describe particles trapped inside of magnetic islands and that are being accelerated during magnetic island contraction and merging (Drake, Swisdak & Fermo Reference Drake, Swisdak and Fermo2013; Zank et al. Reference Zank, le Roux, Webb, Dosch and Khabarova2014).

2.6 Exact equations of anisotropic multi-fluid models

Before we discuss solutions of specific fluid models, it is beneficial to summarize the most general equations of multi-fluid models that were derived with no simplifications at all. By reintroducing the species index $r$ , the equations that were directly obtained by integrating the collisionless Vlasov equation read

(2.89) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{r}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}_{r}\boldsymbol{u}_{r})=0; & \displaystyle\end{eqnarray}$$

(2.90) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}_{r}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r}+\frac{1}{\unicode[STIX]{x1D70C}_{r}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{p}_{r}-\frac{q_{r}}{m_{r}}\left(\boldsymbol{E}+\frac{1}{c}\boldsymbol{u}_{r}\times \boldsymbol{B}\right)=0; & \displaystyle\end{eqnarray}$$

(2.91) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}p_{\Vert r}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(p_{\Vert r}\boldsymbol{u}_{r})+2p_{\Vert r}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}_{r})\boldsymbol{\cdot }\hat{\boldsymbol{b}}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{2}{|\boldsymbol{B}|}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\frac{\text{d}_{r}\boldsymbol{B}}{\text{d}t}+(\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r})^{S}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=0;\end{eqnarray}$$

(2.92) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}p_{\bot r}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(p_{\bot r}\boldsymbol{u}_{r})+p_{\bot r}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{r}-p_{\bot r}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+\frac{1}{2}[\text{Tr}\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}_{r}-\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}_{r})\boldsymbol{\cdot }\hat{\boldsymbol{b}}]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{1}{2}\left[\text{Tr}(\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r})^{S}+\frac{2}{|\boldsymbol{B}|}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\frac{\text{d}_{r}\boldsymbol{B}}{\text{d}t}-(\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r})^{S}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}\right]=0,\end{eqnarray}$$

where the convective derivative $\text{d}_{r}/\text{d}t=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t+\boldsymbol{u}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ . In the above equations, the pressure tensor for each species was just decomposed into gyrotropic contributions $p_{\Vert r}$ , $p_{\bot r}$ and the rest of the pressure tensor (the FLR pressure tensor $\unicode[STIX]{x1D72B}_{r}$ ), according to

(2.93) $$\begin{eqnarray}\boldsymbol{p}_{r}=p_{\Vert r}\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}+p_{\bot r}(\unicode[STIX]{x1D644}-\hat{\boldsymbol{b}}\hat{\boldsymbol{b}})+\unicode[STIX]{x1D72B}_{r},\end{eqnarray}$$

which is a rigorous decomposition not introducing any simplifications. The heat flux $\boldsymbol{q}_{r}$ contributions are here left at the most general level without any simplifications either. The equations are accompanied by Maxwell’s equations, that for now we write down here with no simplifications to emphasize that no Maxwell equations were used in deriving the above system,

(2.94) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{E}=4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{c}=4\unicode[STIX]{x03C0}\mathop{\sum }_{r}q_{r}n_{r};\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{B}=0; & \displaystyle\end{eqnarray}$$

(2.95) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}=-c\unicode[STIX]{x1D735}\times \boldsymbol{E};\quad \boldsymbol{j}=\mathop{\sum }_{r}q_{r}n_{r}\boldsymbol{u}_{r}=\frac{c}{4\unicode[STIX]{x03C0}}\unicode[STIX]{x1D735}\times \boldsymbol{B}-\frac{1}{4\unicode[STIX]{x03C0}}\frac{\unicode[STIX]{x2202}\boldsymbol{E}}{\unicode[STIX]{x2202}t}. & \displaystyle\end{eqnarray}$$

Equations (2.89)–(2.95) represent an exact (even though not closed) kinetic Vlasov–Maxwell system formulated in fluid variables, and the description is valid for any general distribution function and for all considered spatial and temporal scales. All advanced collisionless multi-fluid models have to be based in one way or another on these equations. One can concentrate on the proton dynamics or one can concentrate on the electron dynamics and simplify these equations accordingly for these spatial scales. Naturally, one can consider complicated multi-fluid descriptions composed of many particle species.

For a new reader who just jumped straight to this section, the symmetric operator ‘ $S$ ’ acts on a matrix $\unicode[STIX]{x1D608}_{ij}$ according to $\unicode[STIX]{x1D608}_{ij}^{S}=\unicode[STIX]{x1D608}_{ij}+\unicode[STIX]{x1D608}_{ji}$ . If one does not like the symmetric operator in the expressions above, the symmetric operator is actually not necessary, since

(2.96) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})_{ij}=\unicode[STIX]{x1D6F1}_{ik}\unicode[STIX]{x2202}_{k}u_{j}; & \displaystyle\end{eqnarray}$$

(2.97) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})_{ij}^{S}=\unicode[STIX]{x1D6F1}_{ik}\unicode[STIX]{x2202}_{k}u_{j}+\unicode[STIX]{x1D6F1}_{jk}\unicode[STIX]{x2202}_{k}u_{i}; & \displaystyle\end{eqnarray}$$

(2.98) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{S}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=2\hat{b}_{i}\unicode[STIX]{x1D6F1}_{ik}(\unicode[STIX]{x2202}_{k}u_{j})\hat{b}_{j}=2(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}):\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}=2\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})\boldsymbol{\cdot }\hat{\boldsymbol{b}}; & \displaystyle\end{eqnarray}$$

(2.99) $$\begin{eqnarray}\displaystyle & \displaystyle \text{Tr}(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u})^{S}=2\unicode[STIX]{x1D6F1}_{ik}\unicode[STIX]{x2202}_{k}u_{i}=2\unicode[STIX]{x1D72B}:\unicode[STIX]{x1D735}\boldsymbol{u}=2(\unicode[STIX]{x1D72B}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{\cdot }\boldsymbol{u}, & \displaystyle\end{eqnarray}$$

where we also got rid of the double contractions. Note that $\unicode[STIX]{x1D6F1}_{ik}=\unicode[STIX]{x1D6F1}_{ki}$ and also $p_{ik}=p_{ki}$ .

Equations (2.89)–(2.92) contain two very important catches – the entire heat flux tensor $\boldsymbol{q}_{r}$ and the non-gyrotropic pressure tensor $\unicode[STIX]{x1D72B}_{r}$ are not specified yet. In this moment, the entire class of collisionless fluid models separates to many possible classes and sub-classes, depending on how precisely one wants to evaluate the heat flux tensor $\boldsymbol{q}_{r}$ and the FLR pressure tensor $\unicode[STIX]{x1D72B}_{r}$ . There are several complications how to correctly model these two quantities, but before we discuss this, we want to point out that by using the simple general identities (3.46), (3.47) (that are nothing more than calculating derivatives and using the density equation for each species separately), the above pressure equations can be rewritten to the following form

(2.100) $$\begin{eqnarray}\displaystyle \frac{\text{d}_{r}}{\text{d}t}\left(\frac{p_{\Vert r}|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}_{r}^{3}}\right) & = & \displaystyle \frac{|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}_{r}^{3}}\left\{2p_{\Vert r}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{r}-2p_{\Vert r}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+2\frac{p_{\Vert r}}{|\boldsymbol{B}|}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\frac{\text{d}_{r}\boldsymbol{B}}{\text{d}t}-\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}_{r})\boldsymbol{\cdot }\hat{\boldsymbol{b}}\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\frac{2}{|\boldsymbol{B}|}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\frac{\text{d}_{r}\boldsymbol{B}}{\text{d}t}-(\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r})^{S}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}\right\};\end{eqnarray}$$

(2.101) $$\begin{eqnarray}\displaystyle \frac{\text{d}_{r}}{\text{d}t}\left(\frac{p_{\bot r}}{\unicode[STIX]{x1D70C}_{r}|\boldsymbol{B}|}\right) & = & \displaystyle \frac{1}{\unicode[STIX]{x1D70C}_{r}|\boldsymbol{B}|}\left\{-p_{\bot r}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{r}+p_{\bot r}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r}\boldsymbol{\cdot }\hat{\boldsymbol{b}}-\frac{p_{\bot r}}{|\boldsymbol{B}|}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\frac{\text{d}_{r}\boldsymbol{B}}{\text{d}t}\right.\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{2}[\text{Tr}\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}_{r}-\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}_{r})\boldsymbol{\cdot }\hat{\boldsymbol{b}}]\nonumber\\ \displaystyle & & \displaystyle -\left.\frac{1}{2}\left[\text{Tr}(\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r})^{S}+\frac{2}{|\boldsymbol{B}|}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\frac{\text{d}_{r}\boldsymbol{B}}{\text{d}t}-(\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r})^{S}:\hat{\boldsymbol{b}}\hat{\boldsymbol{b}}\right]\right\}.\end{eqnarray}$$

Again, no simplifications were introduced and no Maxwell equations were used yet. The equations are exact. The equations represent the complicated plasma heating processes that are responsible for anisotropic plasma heating, and that are in general very difficult to classify. The left-hand sides of (2.100)–(2.101) are the familiar CGL equations that represent conservations of the first and second adiabatic invariants, and all the expressions on the right-hand sides break these adiabatic invariants. On the right-hand side, we have the $\unicode[STIX]{x2202}\boldsymbol{B}/\unicode[STIX]{x2202}t$ that couples various species together through Maxwell’s equations, and that also introduces the Hall term responsible for the simplest dispersive effects. The heat flux tensor $\boldsymbol{q}_{r}$ (a tensor of third rank), is decomposed into its gyrotropic and non-gyrotropic (FLR) parts $\boldsymbol{q}_{r}=\boldsymbol{q}_{r}^{\text{g}}+\boldsymbol{q}_{r}^{\text{ng}}$ . At large scales, the gyrotropic heat flux can be viewed as a gateway for the simplest form of collisionless damping mechanism, known as Landau damping, that can be further separated to a ‘pure’ electrostatic Landau damping, and its magnetic analogue, the transit-time damping – see Part 2 of the text, subsection ‘Coulomb force and mirror force (Landau damping and transit time damping)’. Nevertheless, fluid models that contain the gyrotropic heat flux fluctuations, but that do not contain any Landau damping, can be constructed as well (see CGL2 model later in the text). The FLR pressure tensor $\unicode[STIX]{x1D72B}_{r}$ introduces further dispersive effects and therefore also modifies the collisionless damping rates, as well as the heat flux $\boldsymbol{q}_{r}$ modifies the FLR pressure corrections $\unicode[STIX]{x1D72B}_{r}$ . The quantities $\boldsymbol{q}_{r}$ and $\unicode[STIX]{x1D72B}_{r}$ are therefore generally coupled. Importantly, the FLR pressure tensor $\unicode[STIX]{x1D72B}_{r}$ introduces complicated turbulent plasma heating processes, referred to as stochastic heating. Since these FLR stress forces completely disappear at the linear level in the above pressure equations, they can be explored only with fully nonlinear numerical simulations ( $\unicode[STIX]{x1D72B}_{r}$ still enters at the linear level in the momentum equation and modifies the dispersion relations).

From a fluid perspective, there are several major difficulties how to correctly model the quantities $\unicode[STIX]{x1D72B}_{r}$ and $\boldsymbol{q}_{r}$ . The heat flux tensor $\boldsymbol{q}_{r}$ is described by an infinite hierarchy of higher-order fluid moments, and one needs to find an appropriate way, how to close the system. The FLR pressure tensor $\unicode[STIX]{x1D72B}_{r}$ is described by equations that are implicit, and the FLR corrections have to be typically expanded on temporal and spatial scales in order to obtain tractable expressions that can be used for numerical simulations. This expansion will naturally restrict the area of validity of such fluid models to those scales considered. For example, for the proton–electron plasma where the spatial and temporal scales are largely separated because of the mass ratio $m_{p}/m_{e}=1836$ , simple first-order FLR corrections expanded around the proton scales $k\unicode[STIX]{x1D70C}_{i},\unicode[STIX]{x1D714}/\unicode[STIX]{x1D6FA}_{p}$ will long lose their validity at the electron scales. The most complicated Landau fluid models that use linear kinetic theory to evaluate $\unicode[STIX]{x1D72B}$ and that contain the Bessel functions (Passot & Sulem Reference Passot and Sulem2007; Passot et al. Reference Passot, Sulem and Hunana2012; Sulem & Passot Reference Sulem and Passot2015), have technically no restriction for wavenumbers $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , but the first-order frequency restriction is there nevertheless.

2.7 Conservation of energy

The equations (2.100)–(2.101), that correctly split the entire plasma heating into parallel and perpendicular heating, are indeed quite complicated. To gain further insight, let us briefly consider a case in which we are not interested in parallel and perpendicular heating (whose net effect can possibly be zero), and we are only interested in the heating of the entire system. Such formulations are sometimes used to interpret plasma heating in fully kinetic simulations (Yang et al. Reference Yang, Matthaeus, Parashar, Haggerty, Roytershteyn, Daughton, Wan, Shi and Chen2017a ,Reference Yang, Matthaeus, Parashar, Wu, Wan, Shi, Chen, Roytershteyn and Daughton b ). By summing together the pressure equations (2.91), (2.92), we are interested in the evolution equation $\unicode[STIX]{x2202}(p_{\Vert }+2p_{\bot })/\unicode[STIX]{x2202}t$ , or in another words, we are interested in $\unicode[STIX]{x2202}\text{Tr}\,\boldsymbol{p}/\unicode[STIX]{x2202}t$ . Summing the equations together (actually the trace of the entire pressure tensor equation was calculated a few pages back, that is how the $p_{\bot }$ equation was derived) yields

(2.102) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(p_{\Vert r}+2p_{\bot r})+\unicode[STIX]{x1D735}\boldsymbol{\cdot }((p_{\Vert r}+2p_{\bot r})\boldsymbol{u}_{r})+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\text{Tr}\,\boldsymbol{q}_{r})+2((\underbrace{\boldsymbol{p}_{r}^{g}+\unicode[STIX]{x1D72B}_{r}}_{=\boldsymbol{p}_{r}})\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{\cdot }\boldsymbol{u}_{r}=0. & & \displaystyle\end{eqnarray}$$

Note that $\text{Tr}\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}_{r}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\text{Tr}\,\boldsymbol{q}_{r})$ , and also $\text{Tr}\,\unicode[STIX]{x1D72B}_{r}=0$ . By integrating over the entire spatial volume of the plasma, one can define the internal energy (or thermal energy) for each particle species as

(2.103) $$\begin{eqnarray}E_{r}^{\text{int}}=\frac{1}{2}\int \text{Tr}\,\boldsymbol{p}_{r}\,\text{d}^{3}x=\frac{1}{2}\int (p_{\Vert r}+2p_{\bot r})\,\text{d}^{3}x\equiv \frac{1}{2}\langle p_{\Vert r}+2p_{\bot r}\rangle .\end{eqnarray}$$

By considering a special case of periodic boundary conditions, i.e. typical numerical simulations of turbulence in a periodic box, one can use the Gauss–Ostrogradsky theorem (2.9), now in a spatial domain, and for periodic boundary conditions all divergence operators in (2.102) vanish, yielding

(2.104) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}E_{r}^{\text{int}}=-\langle ((\boldsymbol{p}_{r}^{g}+\unicode[STIX]{x1D72B}_{r})\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{\cdot }\boldsymbol{u}_{r}\rangle .\end{eqnarray}$$

Equations (2.102), (2.104) are equivalent to (4) and (7) of Yang et al. (Reference Yang, Matthaeus, Parashar, Wu, Wan, Shi, Chen, Roytershteyn and Daughton2017b ). Here the equations are just written in such a way that one can explicitly see the pressure components $p_{\Vert },p_{\bot },\unicode[STIX]{x1D72B}$ , but the equations are equivalent. Following Yang et al. (Reference Yang, Matthaeus, Parashar, Wu, Wan, Shi, Chen, Roytershteyn and Daughton2017b ), one can also easily derive the evolution equations for the ‘kinetic’ energy and the electromagnetic energy

(2.105) $$\begin{eqnarray}E_{r}^{\text{kin}}=\frac{1}{2}\langle \unicode[STIX]{x1D70C}_{r}|\boldsymbol{u}_{r}|^{2}\rangle ;\quad E^{\text{mag}}=\frac{1}{8\unicode[STIX]{x03C0}}\langle (|\boldsymbol{B}|^{2}+|\boldsymbol{E}|^{2})\rangle .\end{eqnarray}$$

For the kinetic energy, by first applying $\boldsymbol{u}_{r}\boldsymbol{\cdot }$ to the momentum equation (2.90)

(2.106) $$\begin{eqnarray}\boldsymbol{u}_{r}\boldsymbol{\cdot }\left(\unicode[STIX]{x1D70C}_{r}\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{r}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D70C}_{r}\boldsymbol{u}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r}\right)=-\boldsymbol{u}_{r}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{p}_{r})+\underbrace{q_{r}n_{r}\boldsymbol{u}_{r}}_{=\boldsymbol{j}_{r}}\boldsymbol{\cdot }\,\boldsymbol{E},\end{eqnarray}$$

which further implies

(2.107) $$\begin{eqnarray}\displaystyle \frac{1}{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(\unicode[STIX]{x1D70C}|\boldsymbol{u}|^{2}) & = & \displaystyle \frac{1}{2}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}t}|\boldsymbol{u}|^{2}+\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}|\boldsymbol{u}|^{2}}{\unicode[STIX]{x2202}t}\right)=-\frac{1}{2}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}\boldsymbol{u})|\boldsymbol{u}|^{2}+\unicode[STIX]{x1D70C}\boldsymbol{u}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}\nonumber\\ \displaystyle & = & \displaystyle -\frac{1}{2}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}\boldsymbol{u}|\boldsymbol{u}|^{2})+\boldsymbol{u}\boldsymbol{\cdot }\left(\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D70C}\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\right)\nonumber\\ \displaystyle & = & \displaystyle -\frac{1}{2}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}_{r}\boldsymbol{u}_{r}|\boldsymbol{u}_{r}|^{2})-\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{p}_{r}\boldsymbol{u}_{r})+(\boldsymbol{p}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{\cdot }\boldsymbol{u}_{r}+\boldsymbol{j}_{r}\boldsymbol{\cdot }\boldsymbol{E},\end{eqnarray}$$

and for the electromagnetic energy

(2.108) $$\begin{eqnarray}\displaystyle \frac{1}{8\unicode[STIX]{x03C0}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(|\boldsymbol{B}|^{2}+|\boldsymbol{E}|^{2}) & = & \displaystyle \frac{1}{4\unicode[STIX]{x03C0}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\boldsymbol{B}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}+\boldsymbol{E}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}\boldsymbol{E}}{\unicode[STIX]{x2202}t}\right)\nonumber\\ \displaystyle & = & \displaystyle -\frac{c}{4\unicode[STIX]{x03C0}}(\boldsymbol{B}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{E})-\boldsymbol{E}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{B}))-\boldsymbol{j}\boldsymbol{\cdot }\boldsymbol{E}\nonumber\\ \displaystyle & = & \displaystyle -\frac{c}{4\unicode[STIX]{x03C0}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{E}\times \boldsymbol{B})-\boldsymbol{j}\boldsymbol{\cdot }\boldsymbol{E}.\end{eqnarray}$$

Therefore, by integrating over the entire volume and assuming periodic boundary conditions, the total conservation of energy can be expressed as (Yang et al. Reference Yang, Matthaeus, Parashar, Wu, Wan, Shi, Chen, Roytershteyn and Daughton2017b )

(2.109) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}E_{r}^{\text{int}} & = & \displaystyle -\langle (\boldsymbol{p}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{\cdot }\boldsymbol{u}_{r}\rangle ;\end{eqnarray}$$

(2.110) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}E_{r}^{\text{kin}} & = & \displaystyle +\langle (\boldsymbol{p}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{\cdot }\boldsymbol{u}_{r}\rangle +\langle \boldsymbol{j}_{r}\boldsymbol{\cdot }\boldsymbol{E}\rangle ;\end{eqnarray}$$

(2.111) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}E^{\text{mag}} & = & \displaystyle -\langle \boldsymbol{j}\boldsymbol{\cdot }\boldsymbol{E}\rangle ,\end{eqnarray}$$

which beautifully clarifies how the energy can be transferred. Nevertheless, we point out that it is exactly the splitting into the parallel and perpendicular heating that is so complicated, since even with periodic boundary conditions, only a few terms in the pressure equations (2.91), (2.92) vanish. Considering anisotropic heating, and splitting the internal (thermal) energy into parallel and perpendicular parts

(2.112) $$\begin{eqnarray}E_{r}^{\text{int}}=E_{\Vert r}^{\text{int}}+E_{\bot r}^{\text{int}};\quad E_{\Vert r}^{\text{int}}={\textstyle \frac{1}{2}}\langle p_{\Vert r}\rangle ;\quad E_{\bot r}^{\text{int}}=\langle p_{\bot r}\rangle ,\end{eqnarray}$$

the conservation of total energy can be expressed as

(2.113) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}E_{\Vert r}^{\text{int}} & = & \displaystyle -\langle p_{\Vert r}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r}\boldsymbol{\cdot }\hat{\boldsymbol{b}}\rangle -\frac{1}{2}\langle \hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}_{r})\boldsymbol{\cdot }\hat{\boldsymbol{b}}\rangle \nonumber\\ \displaystyle & & \displaystyle +\,\langle \frac{\hat{\boldsymbol{b}}}{|\boldsymbol{B}|}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\frac{\text{d}_{r}\boldsymbol{B}}{\text{d}t}\rangle -\langle \hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r})\boldsymbol{\cdot }\hat{\boldsymbol{b}}\rangle ;\end{eqnarray}$$

(2.114) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}E_{\bot r}^{\text{int}} & = & \displaystyle -\langle p_{\bot r}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{r}\rangle +\langle p_{\bot r}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r}\boldsymbol{\cdot }\hat{\boldsymbol{b}}\rangle +\frac{1}{2}\langle \hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{q}_{r})\boldsymbol{\cdot }\hat{\boldsymbol{b}}\rangle \nonumber\\ \displaystyle & & \displaystyle -\,\langle \frac{\hat{\boldsymbol{b}}}{|\boldsymbol{B}|}\boldsymbol{\cdot }\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\frac{\text{d}_{r}\boldsymbol{B}}{\text{d}t}\rangle +\langle \hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r})\boldsymbol{\cdot }\hat{\boldsymbol{b}}\rangle -\langle (\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{\cdot }\boldsymbol{u}_{r}\rangle ;\end{eqnarray}$$

(2.115) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}E_{r}^{\text{kin}} & = & \displaystyle +\langle (\boldsymbol{p}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{\cdot }\boldsymbol{u}_{r}\rangle +\langle \boldsymbol{j}_{r}\boldsymbol{\cdot }\boldsymbol{E}\rangle ;\end{eqnarray}$$

(2.116) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}E^{\text{mag}} & = & \displaystyle -\langle \boldsymbol{j}\boldsymbol{\cdot }\boldsymbol{E}\rangle ,\end{eqnarray}$$

where

(2.117) $$\begin{eqnarray}\langle (\boldsymbol{p}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{\cdot }\boldsymbol{u}_{r}\rangle =\langle p_{\Vert r}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r}\boldsymbol{\cdot }\hat{\boldsymbol{b}}\rangle -\langle p_{\bot r}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{r}\boldsymbol{\cdot }\hat{\boldsymbol{b}}\rangle +\langle p_{\bot r}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{r}\rangle +\langle (\unicode[STIX]{x1D72B}_{r}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{\cdot }\boldsymbol{u}_{r}\rangle .\end{eqnarray}$$

The anisotropic plasma heating is obviously a very complicated process.

3.1 Normalized equations and definitions

As with MHD and Hall-MHD, it is often useful to work with normalized equations. The density, velocity, magnetic field and pressure are normalized with respect to $\unicode[STIX]{x1D70C}_{0},u_{0},B_{0},p_{\Vert }^{(0)}$ . The length is normalized with respect to (for now) arbitrary $x_{0}$ and the time with respect to $t_{0}=x_{0}/u_{0}$ . The normalized quantities are then

(3.23) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D70C}}=\frac{\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70C}_{0}};\quad \widetilde{\boldsymbol{u}}=\frac{\boldsymbol{u}}{u_{0}};\quad \widetilde{\boldsymbol{B}}=\frac{\boldsymbol{B}}{B_{0}};\quad \widetilde{p}_{\Vert }=\frac{p_{\Vert }}{p_{\Vert }^{(0)}};\quad \widetilde{p}_{\bot }=\frac{p_{\bot }}{p_{\Vert }^{(0)}}; & & \displaystyle\end{eqnarray}$$

(3.24) $$\begin{eqnarray}\displaystyle \widetilde{\boldsymbol{x}}=\frac{\boldsymbol{x}}{x_{0}};\quad \widetilde{\unicode[STIX]{x1D735}}=\unicode[STIX]{x1D735}x_{0};\quad \widetilde{t}=t\frac{u_{0}}{x_{0}};\quad \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\widetilde{t}}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\frac{x_{0}}{u_{0}}. & & \displaystyle\end{eqnarray}$$

Obviously $\widetilde{n}=\widetilde{\unicode[STIX]{x1D70C}}$ . After dropping the tilde, the normalization yields the same density and pressure equations, whereas the normalized momentum and induction equation become

(3.25) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}_{p}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{p}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{p}+\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}u_{0}^{2}}\frac{1}{\unicode[STIX]{x1D70C}_{p}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{p}_{p}+\boldsymbol{p}_{e})-\frac{V_{A}^{2}}{u_{0}^{2}}\frac{1}{\unicode[STIX]{x1D70C}_{p}}(\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}=0; & \displaystyle\end{eqnarray}$$

(3.26) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times (\boldsymbol{u}\times \boldsymbol{B})-\frac{V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}u_{0}x_{0}}\unicode[STIX]{x1D735}\times \left[\frac{1}{n}(\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}\right]+\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D6FA}_{p}\unicode[STIX]{x1D70C}_{0}u_{0}x_{0}}\unicode[STIX]{x1D735}\times \left[\frac{1}{n}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{p}_{e}\right], & \displaystyle\end{eqnarray}$$

where the Alfvén speed $V_{A}=B_{0}/\sqrt{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{0}}$ . The momentum equation implies that it is beneficial to choose the (so far unspecified) normalizing velocity $u_{0}$ to be the Alfvén speed $u_{0}=V_{A}$ . If the second and third terms are neglected in the induction equation (i.e. when the usual MHD induction equation is used), the CGL system is independent of length scale $x_{0}$ , similarly to the MHD system. If the Hall term is considered, it is beneficial to choose the (so far unspecified) normalizing length scale $x_{0}$ to be the ion inertial length

(3.27) $$\begin{eqnarray}d_{i}=\frac{V_{A}}{\unicode[STIX]{x1D6FA}_{p}}.\end{eqnarray}$$

Therefore by choosing $x_{0}=d_{i}$ , the normalizations that we use in all fluid models in this paper read

(3.28) $$\begin{eqnarray}\displaystyle \widetilde{\boldsymbol{u}}=\frac{\boldsymbol{u}}{V_{A}};\quad \widetilde{\boldsymbol{x}}=\frac{\boldsymbol{x}}{d_{i}};\quad \widetilde{t}=t\frac{V_{A}}{d_{i}}=t\unicode[STIX]{x1D6FA}_{p}. & & \displaystyle\end{eqnarray}$$

The normalization also has the advantage that, when transferred to Fourier space, one obtains the dispersion relations for normalized wavenumber $\widetilde{k}$ and frequency $\widetilde{\unicode[STIX]{x1D714}}$ as

(3.29) $$\begin{eqnarray}\widetilde{k}=\frac{kV_{A}}{\unicode[STIX]{x1D6FA}_{p}},\quad \widetilde{\unicode[STIX]{x1D714}}=\frac{\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D6FA}_{p}}.\end{eqnarray}$$

It is useful to define the parallel and perpendicular thermal speeds

(3.30) $$\begin{eqnarray}v_{\text{th}\Vert }=\sqrt{\frac{2T_{\Vert }^{(0)}}{m_{p}}};\quad v_{\text{th}\bot }=\sqrt{\frac{2T_{\bot }^{(0)}}{m_{p}}},\end{eqnarray}$$

and the definition of parallel and perpendicular temperatures are $T_{\Vert }=p_{\Vert }/n$ and $T_{\bot }=p_{\bot }/n$ . We use the usual convention with the Boltzmann constant $k_{B}=1$ .Footnote ³ The parallel and perpendicular plasma beta are defined according to

(3.31) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\Vert }=\frac{v_{\text{th}\Vert }^{2}}{V_{A}^{2}}=\frac{2p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}V_{A}^{2}}=\frac{p_{\Vert }^{(0)}}{B_{0}^{2}/(8\unicode[STIX]{x03C0})};\quad \unicode[STIX]{x1D6FD}_{\bot }=\frac{v_{\text{th}\bot }^{2}}{V_{A}^{2}}=\frac{2p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}V_{A}^{2}}=\frac{p_{\bot }^{(0)}}{B_{0}^{2}/(8\unicode[STIX]{x03C0})}=\unicode[STIX]{x1D6FD}_{\Vert }\frac{T_{\bot }^{(0)}}{T_{\Vert }^{(0)}},\end{eqnarray}$$

and later we will often use the abbreviation $a_{p}$ for the proton temperature anisotropy ratio

(3.32) $$\begin{eqnarray}a_{p}=\frac{T_{\bot }^{(0)}}{T_{\Vert }^{(0)}}=\frac{p_{\bot }^{(0)}}{p_{\Vert }^{(0)}}.\end{eqnarray}$$

The normalized momentum and induction equations therefore read (tilde are dropped)

(3.33) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}_{p}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{p}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{p}+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}\frac{1}{\unicode[STIX]{x1D70C}_{p}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\boldsymbol{p}_{p}+\boldsymbol{p}_{e})-\frac{1}{\unicode[STIX]{x1D70C}_{p}}(\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}=0; & \displaystyle\end{eqnarray}$$

(3.34) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times (\boldsymbol{u}\times \boldsymbol{B})-\unicode[STIX]{x1D735}\times \left[\frac{1}{n}(\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}\right]+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}\unicode[STIX]{x1D735}\times \left[\frac{1}{n}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{p}_{e}\right]. & \displaystyle\end{eqnarray}$$

Another useful quantity that we will use later in the text is the proton Larmor radius (the gyroradius) $\unicode[STIX]{x1D70C}_{i}$ , defined according to

(3.35) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{i}=\frac{v_{\text{th}\bot }}{\unicode[STIX]{x1D6FA}_{p}}.\end{eqnarray}$$

We note that normalizations do not have to be done with respect to mean values, and normalizations with respect to fluctuating (turbulent) quantities are used for example in the nearly incompressible models of Zank et al. (Reference Zank, Adhikari, Hunana, Shiota, Bruno and Telloni2017) and Zank & Matthaeus (Reference Zank and Matthaeus1993).

3.2 Classical CGL model with cold electrons

Here we want to consider the simplest possible CGL model for the proton species. We assume the electrons to be massless and cold ( $\boldsymbol{p}_{e}=0$ ), and we also neglect the Hall term in the induction equation. Since only proton species are present, for simplicity we can drop the proton index ‘ $p$ ’. Let us work in physical units for a moment. The classical CGL model with cold electrons therefore reads (written in physical units)

(3.36) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D70C}\boldsymbol{u})=0; & \displaystyle\end{eqnarray}$$

(3.37) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}+\frac{1}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{p}-\frac{1}{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}=0; & \displaystyle\end{eqnarray}$$

(3.38) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}p_{\Vert }}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(p_{\Vert }\boldsymbol{u})+2p_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}=0; & \displaystyle\end{eqnarray}$$

(3.39) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}p_{\bot }}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(p_{\bot }\boldsymbol{u})+p_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}-p_{\bot }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}=0; & \displaystyle\end{eqnarray}$$

(3.40) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times (\boldsymbol{u}\times \boldsymbol{B}). & \displaystyle\end{eqnarray}$$

Similarly to the usual MHD model, the CGL equations are scale invariant and do not contain any information about the physical length scale. A length scale is introduced by considering the Hall term in the induction equation, and we discuss the Hall-CGL model in the next section. Also, the absence of the Hall term allows the CGL pressure equations to be rewritten into ‘conservative’ form as

(3.41) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\left(\frac{p_{\Vert }|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{3}}\right)=0;\quad \frac{\text{d}}{\text{d}t}\left(\frac{p_{\bot }}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\right)=0,\end{eqnarray}$$

where $\text{d}/\text{d}t$ is the convective derivative. This formulation is the most often cited form of the CGL pressure equations (often the $|\boldsymbol{B}|$ is abbreviated only as $B$ , which can lead to confusion on how to correctly calculate the derivatives). We want to show the equivalence of (3.41) and (3.38), (3.39). We need the following identity,

(3.42) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}|\boldsymbol{B}|=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}(B_{x}^{2}+B_{y}^{2}+B_{z}^{2})^{1/2}=\frac{1}{|\boldsymbol{B}|}\boldsymbol{B}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}=\hat{\boldsymbol{b}}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}, & & \displaystyle\end{eqnarray}$$

and the result is of course a scalar. Similarly for the spatial and convective derivative

(3.43) $$\begin{eqnarray}\unicode[STIX]{x2202}_{i}|\boldsymbol{B}|=\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x2202}_{i}\boldsymbol{B});\quad \frac{\text{d}}{\text{d}t}|\boldsymbol{B}|=\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+u_{i}\unicode[STIX]{x2202}_{i}\right)|\boldsymbol{B}|=\hat{\boldsymbol{b}}\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{B}}{\text{d}t}.\end{eqnarray}$$

Later we will need

(3.44) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}|\boldsymbol{B}|^{2}=\frac{\text{d}}{\text{d}t}(B_{x}^{2}+B_{y}^{2}+B_{z}^{2})=2\boldsymbol{B}\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{B}}{\text{d}t}.\end{eqnarray}$$

Now we can directly calculate

(3.45) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{p_{\bot }}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\right) & = & \displaystyle \frac{1}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\frac{\text{d}p_{\bot }}{\text{d}t}-\frac{p_{\bot }}{|\boldsymbol{B}|\unicode[STIX]{x1D70C}^{2}}\frac{\text{d}\unicode[STIX]{x1D70C}}{\text{d}t}-\frac{p_{\bot }}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|^{2}}\frac{\text{d}|\boldsymbol{B}|}{\text{d}t}\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\left(\frac{\text{d}p_{\bot }}{\text{d}t}-\frac{p_{\bot }}{\unicode[STIX]{x1D70C}}\frac{\text{d}\unicode[STIX]{x1D70C}}{\text{d}t}-\frac{p_{\bot }}{|\boldsymbol{B}|}\frac{\text{d}|\boldsymbol{B}|}{\text{d}t}\right),\end{eqnarray}$$

and, by using the density equation $\text{d}\unicode[STIX]{x1D70C}/\text{d}t=-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}$ and the identity (3.43), we obtain

(3.46) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{p_{\bot }}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\right)=\frac{1}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\left(\frac{\text{d}p_{\bot }}{\text{d}t}+p_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}-\frac{p_{\bot }}{|\boldsymbol{B}|}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{B}}{\text{d}t}\right). & & \displaystyle\end{eqnarray}$$

The above equation is completely general since we did not use the induction equation so far and we will use this equation in the next section when we consider the Hall-CGL model. In a similar way one derives a completely general identity

(3.47) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{p_{\Vert }|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{3}}\right) & = & \displaystyle \frac{|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{3}}\left(\frac{\text{d}p_{\Vert }}{\text{d}t}+\frac{p_{\Vert }}{|\boldsymbol{B}|^{2}}\frac{\text{d}|\boldsymbol{B}|^{2}}{\text{d}t}-3\frac{p_{\Vert }}{\unicode[STIX]{x1D70C}}\frac{\text{d}\unicode[STIX]{x1D70C}}{\text{d}t}\right)\nonumber\\ \displaystyle & = & \displaystyle \frac{|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{3}}\left(\frac{\text{d}p_{\Vert }}{\text{d}t}+3p_{\Vert }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}+\frac{2p_{\Vert }}{|\boldsymbol{B}|}\hat{\boldsymbol{b}}\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{B}}{\text{d}t}\right).\end{eqnarray}$$

Now, we use the induction equation

(3.48) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times (\boldsymbol{u}\times \boldsymbol{B})=\boldsymbol{u}\underbrace{\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{B}}_{=0}-\boldsymbol{B}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}+\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}-\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{B},\end{eqnarray}$$

that allows us to calculate

(3.49) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}\boldsymbol{B}}{\text{d}t}=\frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{B}=-\boldsymbol{B}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}+\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}; & \displaystyle\end{eqnarray}$$

(3.50) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\hat{\boldsymbol{b}}}{|\boldsymbol{B}|}\boldsymbol{\cdot }\frac{\text{d}\boldsymbol{B}}{\text{d}t}=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}+\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}, & \displaystyle\end{eqnarray}$$

and which, when used in (3.47)–(3.46) yields the final result

(3.51) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{p_{\Vert }|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{3}}\right)=\frac{|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{3}}\left(\frac{\text{d}p_{\Vert }}{\text{d}t}+p_{\Vert }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}+2p_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}\right)=0; & \displaystyle\end{eqnarray}$$

(3.52) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{p_{\bot }}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\right)=\frac{1}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\left(\frac{\text{d}p_{\bot }}{\text{d}t}+2p_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}-p_{\bot }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}\right)=0. & \displaystyle\end{eqnarray}$$

Equations (3.51)–(3.52) verify that the CGL pressure equations (3.41) written in the ‘conservative’ form and the directly derived CGL equations (3.38)–(3.39) are equivalent. We will see that the right-hand side of the usual CGL equations (3.41) is non-zero if the Hall term is considered, and that it is further modified by the heat flux contributions and by the FLR stress forces.

3.3 Linearized CGL equations

To obtain the dispersion relations, the equations need to be linearized and transformed to Fourier space. Equations (3.36)–(3.40) are linearized with respect to mean values $\unicode[STIX]{x1D70C}_{0};\boldsymbol{u}^{(0)}\equiv \langle \boldsymbol{u}\rangle =0;p_{\Vert \bot }^{(0)};\boldsymbol{B}_{0}=(0,0,B_{0});\hat{\boldsymbol{b}}_{0}=(0,0,1)$ where the magnetic field $B_{0}$ is assumed to be in the $z$ -direction. We assume that the mean value of the velocity is zero (the zero mean velocity value $\boldsymbol{u}^{(0)}=0$ should not be confused with the normalizing velocity as in (3.23), which is later chosen to be $V_{A}$ ). The terms $\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}$ are linearized as $\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}\overset{\text{lin}}{=}\unicode[STIX]{x2202}_{z}u_{z}$ and the linearized system reads

(3.53) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D70C}_{0}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0; & \displaystyle\end{eqnarray}$$

(3.54) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\frac{1}{\unicode[STIX]{x1D70C}_{0}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{p}-\frac{1}{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{0}}(\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}_{0}=0; & \displaystyle\end{eqnarray}$$

(3.55) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}p_{\Vert }}{\unicode[STIX]{x2202}t}+p_{\Vert }^{(0)}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}+2p_{\Vert }^{(0)}\unicode[STIX]{x2202}_{z}u_{z}=0; & \displaystyle\end{eqnarray}$$

(3.56) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}p_{\bot }}{\unicode[STIX]{x2202}t}+2p_{\bot }^{(0)}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}-p_{\bot }^{(0)}\unicode[STIX]{x2202}_{z}u_{z}=0; & \displaystyle\end{eqnarray}$$

(3.57) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times (\boldsymbol{u}\times \boldsymbol{B}_{0}). & \displaystyle\end{eqnarray}$$

The divergence of the pressure tensor is

(3.58) $$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{p})_{i}=\unicode[STIX]{x2202}_{j}p_{ji}=\hat{b}_{i}\hat{b}_{j}\unicode[STIX]{x2202}_{j}p_{\Vert }+(\unicode[STIX]{x1D6FF}_{ij}-\hat{b}_{i}\hat{b}_{j})\unicode[STIX]{x2202}_{j}p_{\bot }+(p_{\Vert }-p_{\bot })(\hat{b}_{i}\unicode[STIX]{x2202}_{j}\hat{b}_{j}+\hat{b}_{j}\unicode[STIX]{x2202}_{j}\hat{b}_{i}), & & \displaystyle\end{eqnarray}$$

and the first step of linearization yields, by components,

(3.59) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{p})_{x}\overset{\text{lin}}{=}\unicode[STIX]{x2202}_{x}p_{\bot }+(p_{\Vert }^{(0)}-p_{\bot }^{(0)})\unicode[STIX]{x2202}_{z}\hat{b}_{x}; & \displaystyle\end{eqnarray}$$

(3.60) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{p})_{y}\overset{\text{lin}}{=}\unicode[STIX]{x2202}_{y}p_{\bot }+(p_{\Vert }^{(0)}-p_{\bot }^{(0)})\unicode[STIX]{x2202}_{z}\hat{b}_{y}; & \displaystyle\end{eqnarray}$$

(3.61) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{p})_{z}\overset{\text{lin}}{=}\unicode[STIX]{x2202}_{z}p_{\Vert }+(p_{\Vert }^{(0)}-p_{\bot }^{(0)})(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\hat{\boldsymbol{b}}+\unicode[STIX]{x2202}_{z}\hat{b}_{z}), & \displaystyle\end{eqnarray}$$

which can be conveniently written in matrix notation as

(3.62) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{p}\overset{\text{lin}}{=}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\begin{array}{@{}ccc@{}}p_{\bot } & 0 & 0\\ 0 & p_{\bot } & 0\\ 0 & 0 & p_{\Vert }\end{array}\right)+(p_{\Vert }^{(0)}-p_{\bot }^{(0)})\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left.\left(\begin{array}{@{}ccc@{}}0 & 0 & \hat{b}_{x}\\ 0 & 0 & \hat{b}_{y}\\ \hat{b}_{x} & \hat{b}_{y} & 2\hat{b}_{z}\end{array}\right)\right|_{\text{lin}}. & & \displaystyle\end{eqnarray}$$

Notice that, since $\unicode[STIX]{x2202}_{i}|\boldsymbol{B}|=\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x2202}_{i}\boldsymbol{B})$ , the derivatives of unit vectors are calculated according to

(3.63) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{i}\hat{\boldsymbol{b}}=\unicode[STIX]{x2202}_{i}\frac{\boldsymbol{B}}{|\boldsymbol{B}|}=\frac{1}{|\boldsymbol{B}|}\left[\unicode[STIX]{x2202}_{i}\boldsymbol{B}-\frac{\boldsymbol{B}}{|\boldsymbol{B}|}\unicode[STIX]{x2202}_{i}|\boldsymbol{B}|\right]=\frac{1}{|\boldsymbol{B}|}[\unicode[STIX]{x2202}_{i}\boldsymbol{B}-\hat{\boldsymbol{b}}(\hat{\boldsymbol{b}}\boldsymbol{\cdot }(\unicode[STIX]{x2202}_{i}\boldsymbol{B}))], & & \displaystyle\end{eqnarray}$$

and in the linear approximation one has

(3.64) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{i}\hat{\boldsymbol{b}}\overset{\text{lin}}{=}\frac{1}{B_{0}}[\unicode[STIX]{x2202}_{i}\boldsymbol{B}-\hat{\boldsymbol{b}}_{0}\unicode[STIX]{x2202}_{i}B_{z}], & & \displaystyle\end{eqnarray}$$

which by components reads

(3.65) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{i}\hat{b}_{x}\overset{\text{lin}}{=}\frac{1}{B_{0}}\unicode[STIX]{x2202}_{i}B_{x};\quad \unicode[STIX]{x2202}_{i}\hat{b}_{y}\overset{\text{lin}}{=}\frac{1}{B_{0}}\unicode[STIX]{x2202}_{i}B_{y};\quad \unicode[STIX]{x2202}_{i}\hat{b}_{z}\overset{\text{lin}}{=}\frac{1}{B_{0}}(\unicode[STIX]{x2202}_{i}B_{z}-\unicode[STIX]{x2202}_{z}B_{z}), & & \displaystyle\end{eqnarray}$$

where, importantly, $\unicode[STIX]{x2202}_{z}\hat{b}_{z}\overset{\text{lin}}{=}0$ . The divergence of the pressure tensor in the liner approximation is therefore expressed as

(3.66) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{p}\overset{\text{lin}}{=}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\begin{array}{@{}ccc@{}}p_{\bot } & 0 & 0\\ 0 & p_{\bot } & 0\\ 0 & 0 & p_{\Vert }\end{array}\right)+\frac{1}{B_{0}}(p_{\Vert }^{(0)}-p_{\bot }^{(0)})\unicode[STIX]{x1D735}\boldsymbol{\cdot }\left(\begin{array}{@{}ccc@{}}0 & 0 & B_{x}\\ 0 & 0 & B_{y}\\ B_{x} & B_{y} & 0\end{array}\right), & & \displaystyle\end{eqnarray}$$

where the second term contributes if the temperature anisotropy is present, i.e. when $p_{\Vert }^{(0)}\neq p_{\bot }^{(0)}$ . The vectors $(\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}_{0}$ and $\unicode[STIX]{x1D735}\times (\boldsymbol{u}\times \boldsymbol{B}_{0})$ are straightforward to calculate and are

(3.67) $$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}_{0}=B_{0}\left(\begin{array}{@{}c@{}}-\unicode[STIX]{x2202}_{x}B_{z}+\unicode[STIX]{x2202}_{z}B_{x}\\ -\unicode[STIX]{x2202}_{y}B_{z}+\unicode[STIX]{x2202}_{z}B_{y}\\ 0\end{array}\right);\quad \unicode[STIX]{x1D735}\times (\boldsymbol{u}\times \boldsymbol{B}_{0})=B_{0}\left(\begin{array}{@{}c@{}}\unicode[STIX]{x2202}_{z}u_{x}\\ \unicode[STIX]{x2202}_{z}u_{y}\\ -\unicode[STIX]{x2202}_{x}u_{x}-\unicode[STIX]{x2202}_{y}u_{y}\end{array}\right). & & \displaystyle\end{eqnarray}$$

The entire set of linearized CGL equations reads

(3.68) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D70C}_{0}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0; & \displaystyle\end{eqnarray}$$

(3.69) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}u_{x}}{\unicode[STIX]{x2202}t}+\frac{1}{\unicode[STIX]{x1D70C}_{0}}\unicode[STIX]{x2202}_{x}p_{\bot }+\frac{1}{B_{0}\unicode[STIX]{x1D70C}_{0}}(p_{\Vert }^{(0)}-p_{\bot }^{(0)})\unicode[STIX]{x2202}_{z}B_{x}-\frac{B_{0}}{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{0}}(-\unicode[STIX]{x2202}_{x}B_{z}+\unicode[STIX]{x2202}_{z}B_{x})=0; & \displaystyle\end{eqnarray}$$

(3.70) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}u_{y}}{\unicode[STIX]{x2202}t}+\frac{1}{\unicode[STIX]{x1D70C}_{0}}\unicode[STIX]{x2202}_{y}p_{\bot }+\frac{1}{B_{0}\unicode[STIX]{x1D70C}_{0}}(p_{\Vert }^{(0)}-p_{\bot }^{(0)})\unicode[STIX]{x2202}_{z}B_{y}-\frac{B_{0}}{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{0}}(-\unicode[STIX]{x2202}_{y}B_{z}+\unicode[STIX]{x2202}_{z}B_{y})=0; & \displaystyle\end{eqnarray}$$

(3.71) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}u_{z}}{\unicode[STIX]{x2202}t}+\frac{1}{\unicode[STIX]{x1D70C}_{0}}\unicode[STIX]{x2202}_{z}p_{\Vert }+\frac{1}{B_{0}\unicode[STIX]{x1D70C}_{0}}(p_{\Vert }^{(0)}-p_{\bot }^{(0)})(\unicode[STIX]{x2202}_{x}B_{x}+\unicode[STIX]{x2202}_{y}B_{y})=0; & \displaystyle\end{eqnarray}$$

(3.72) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}B_{x}}{\unicode[STIX]{x2202}t}=B_{0}\unicode[STIX]{x2202}_{z}u_{x};\quad \frac{\unicode[STIX]{x2202}B_{y}}{\unicode[STIX]{x2202}t}=B_{0}\unicode[STIX]{x2202}_{z}u_{y};\quad \frac{\unicode[STIX]{x2202}B_{z}}{\unicode[STIX]{x2202}t}=-B_{0}\unicode[STIX]{x2202}_{x}u_{x}-B_{0}\unicode[STIX]{x2202}_{y}u_{y}; & \displaystyle\end{eqnarray}$$

(3.73) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}p_{\Vert }}{\unicode[STIX]{x2202}t}+p_{\Vert }^{(0)}(\unicode[STIX]{x2202}_{x}u_{x}+\unicode[STIX]{x2202}_{y}u_{y})+3p_{\Vert }^{(0)}\unicode[STIX]{x2202}_{z}u_{z}=0; & \displaystyle\end{eqnarray}$$

(3.74) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}p_{\bot }}{\unicode[STIX]{x2202}t}+2p_{\bot }^{(0)}(\unicode[STIX]{x2202}_{x}u_{x}+\unicode[STIX]{x2202}_{y}u_{y})+p_{\bot }^{(0)}\unicode[STIX]{x2202}_{z}u_{z}=0. & \displaystyle\end{eqnarray}$$

Additionally, the 3 components of the induction equation are not independent, but satisfy the $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{B}=0$ constraint. In order to find the linear dispersion relations, without any loss of generality, we consider wave propagation in the $x$ – $z$ plane and assume that there is no variation with respect to the $y$ -coordinate (i.e. one assumes that all expressions containing $\unicode[STIX]{x2202}_{y}$ are zero; alternatively, one can consider propagation in the $y$ – $z$ plane, where all expressions containing $\unicode[STIX]{x2202}_{x}$ would be zero). By exploring the above system, it is noteworthy that the density equation does not play any role in determining the dispersion relation, since no other equation contains the variable $\unicode[STIX]{x1D70C}$ .

3.4 Alfvén mode

On considering (3.68)–(3.74) written in the $x$ – $z$ plane with $\unicode[STIX]{x2202}_{y}=0$ , it is apparent that the equations for $u_{y}$ and $B_{y}$ decouple from the entire system and applying $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t$ to the equation for $u_{y}$ yields

(3.75) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}^{2}u_{y}}{\unicode[STIX]{x2202}t^{2}}-\frac{1}{\unicode[STIX]{x1D70C}_{0}}\left(\frac{B_{0}^{2}}{4\unicode[STIX]{x03C0}}-p_{\Vert }^{(0)}+p_{\bot }^{(0)}\right)\unicode[STIX]{x2202}_{z}^{2}u_{y}=0, & & \displaystyle\end{eqnarray}$$

which is a wave equation describing the propagation of (generally oblique) Alfvén waves in the CGL description. The same wave equation can be obtained for the component $B_{y}$ . For isotropic mean pressures $p_{\Vert }^{(0)}=p_{\bot }^{(0)}$ the equation is equivalent to the MHD description with the usual Alfvén speed $V_{A}=B_{0}/(4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{0})^{1/2}$ . Since

(3.76) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}=\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}V_{A}^{2}};\quad a_{p}\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}=\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}V_{A}^{2}}, & & \displaystyle\end{eqnarray}$$

where $a_{p}$ is the temperature ratio (3.32), the Alfvén wave equation can be rewritten as

(3.77) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}^{2}u_{y}}{\unicode[STIX]{x2202}t^{2}}-V_{A}^{2}\left[1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)\right]\unicode[STIX]{x2202}_{z}^{2}u_{y}=0. & & \displaystyle\end{eqnarray}$$

Considering a wave propagating obliquely in the $x$ – $z$ plane with the wavevector $\boldsymbol{k}=(k_{\bot },0,k_{\Vert })$ at an angle $\unicode[STIX]{x1D703}$ with respect to the $z$ -direction (the $B_{0}$ direction), the parallel and perpendicular wavenumbers are defined as

(3.78) $$\begin{eqnarray}k_{\Vert }=k\cos \unicode[STIX]{x1D703};\quad k_{\bot }=k\sin \unicode[STIX]{x1D703},\end{eqnarray}$$

and the transformation to Fourier space is performed according to (see appendix A)

(3.79) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\rightarrow -\text{i}\unicode[STIX]{x1D714};\quad \unicode[STIX]{x2202}_{z}\rightarrow \text{i}k_{\Vert };\quad \unicode[STIX]{x2202}_{x}\rightarrow \text{i}k_{\bot }.\end{eqnarray}$$

The wave equation (3.77) is then

(3.80) $$\begin{eqnarray}\unicode[STIX]{x1D714}^{2}u_{y}-V_{A}^{2}\left[1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)\right]k^{2}\cos ^{2}\unicode[STIX]{x1D703}u_{y}=0,\end{eqnarray}$$

which yields the dispersion relation for the Alfvén waves

(3.81) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}=\pm k\cos \unicode[STIX]{x1D703}V_{A}\sqrt{1+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FD}_{\Vert }(a_{p}-1)}=\pm k_{\Vert }V_{A}\sqrt{1+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FD}_{\Vert }(a_{p}-1)}. & & \displaystyle\end{eqnarray}$$

It is important to emphasize that, similarly to MHD, the Alfvén mode propagates in all the oblique directions, with a frequency $\unicode[STIX]{x1D714}$ that is proportional to the projection of the wavenumber $\boldsymbol{k}$ in the direction of $B_{0}$ , i.e. $k_{\Vert }=k\cos \unicode[STIX]{x1D703}$ . If the expression under the square root in (3.81) becomes negative, i.e. if $1+\unicode[STIX]{x1D6FD}_{\Vert }(a_{p}-1)/2<0$ , the frequency $\unicode[STIX]{x1D714}$ is imaginary and the Alfvén mode becomes unstable. This instability, that affects the obliquely propagating Alfvén mode, is known as the oblique firehose instability. The necessary (but not sufficient) condition for the instability to develop is $a_{p}=T_{\bot }^{(0)}/T_{\Vert }^{(0)}<1$ . The firehose instability criterion can be rewritten in many forms, the most common being

(3.82) $$\begin{eqnarray}1-\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}+\frac{1}{2}\unicode[STIX]{x1D6FD}_{\Vert }a_{p}<0;\quad \frac{T_{\bot }^{(0)}}{T_{\Vert }^{(0)}}+\frac{2}{\unicode[STIX]{x1D6FD}_{\Vert }}-1<0;\quad \unicode[STIX]{x1D6FD}_{\Vert }-\unicode[STIX]{x1D6FD}_{\bot }>2;\quad p_{\Vert }^{(0)}-p_{\bot }^{(0)}>\frac{B_{0}^{2}}{4\unicode[STIX]{x03C0}}.\end{eqnarray}$$

The first expression directly implies that, if $\unicode[STIX]{x1D6FD}_{\Vert }\leqslant 2$ , no firehose instability can exist. The firehose instability therefore exists only for a relatively high plasma beta $\unicode[STIX]{x1D6FD}_{\Vert }>2$ and only if the parallel temperature is higher than the perpendicular temperature. Importantly, the firehose instability criterion in the CGL model is equivalent to the one found from linear kinetic theory in the long-wavelength limit (for example, Rosenbluth (Reference Rosenbluth1956), Chandrasekhar, Kaufman & Watson (Reference Chandrasekhar, Kaufman and Watson1958), Parker (Reference Parker1958a ) and Vedenov & Sagdeev (Reference Vedenov and Sagdeev1958)). Alfvén wave propagation only affects components $u_{y}$ and $B_{y}$ which are decoupled from the rest of the system, and yields the other (eigenvector) components $u_{x},u_{z},B_{x},B_{z}$ as zero. The $u_{y}$ component does not enter the density equation and Alfvén mode fluctuations therefore do not produce any density fluctuations or pressure fluctuations. The Alfvén mode is therefore incompressible. It is also useful to define the magnetic compressibility $\unicode[STIX]{x1D712}_{b}(k_{\Vert },k_{\bot })$ in Fourier space, that measures the relative ratio of magnetic energy in the parallel component versus the total magnetic energy

(3.83) $$\begin{eqnarray}\unicode[STIX]{x1D712}_{b}(k_{\Vert },k_{\bot })=\frac{|B_{z}|^{2}}{|B_{x}|^{2}+|B_{y}|^{2}+|B_{z}|^{2}}.\end{eqnarray}$$

Since the Alfvén mode produces only $B_{y}$ fluctuations, with $B_{x}=B_{z}=0$ , the magnetic compressibility for this mode is

(3.84) $$\begin{eqnarray}\unicode[STIX]{x1D712}_{b}(k_{\Vert },k_{\bot })=0,\end{eqnarray}$$

for all the propagation directions. Sometimes the magnetic compressibility is defined as a ratio of parallel and perpendicular energies $|B_{z}|^{2}/(|B_{x}|^{2}+|B_{y}|^{2})$ , this quantity however has a slight disadvantage in that the values can become very large (and actually equal to infinity), which makes plotting of this quantity problematic. In contrast, the magnetic compressibility as defined in (3.83), has a nicely bounded range of values between 0 and 1. A similar quantity is defined for the ratio of the energy in velocity fluctuations

(3.85) $$\begin{eqnarray}\unicode[STIX]{x1D712}_{u}(k_{\Vert },k_{\bot })=\frac{|u_{z}|^{2}}{|u_{x}|^{2}+|u_{y}|^{2}+|u_{z}|^{2}},\end{eqnarray}$$

and the Alfvén mode has $\unicode[STIX]{x1D712}_{u}=0$ for all propagation directions, since only the perpendicular velocity component $u_{y}$ is non-zero.

3.5 Slow and fast modes

By applying $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t$ to the equations for $\unicode[STIX]{x2202}u_{x}/\unicode[STIX]{x2202}t$ and $\unicode[STIX]{x2202}u_{z}/\unicode[STIX]{x2202}t$ and using the evolution equations for $B_{x},B_{z},p_{\Vert },p_{\bot }$ one obtains

(3.86) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}^{2}u_{x}}{\unicode[STIX]{x2202}t^{2}}-\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}(2\unicode[STIX]{x2202}_{x}^{2}u_{x}+\unicode[STIX]{x2202}_{x}\unicode[STIX]{x2202}_{z}u_{z})+\left(\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}-\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}-V_{A}^{2}\right)\unicode[STIX]{x2202}_{z}^{2}u_{x}-V_{A}^{2}\unicode[STIX]{x2202}_{x}^{2}u_{x}=0; & \displaystyle\end{eqnarray}$$

(3.87) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}^{2}u_{z}}{\unicode[STIX]{x2202}t^{2}}-\frac{3p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}\unicode[STIX]{x2202}_{z}^{2}u_{z}-\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}\unicode[STIX]{x2202}_{x}\unicode[STIX]{x2202}_{z}u_{x}=0. & \displaystyle\end{eqnarray}$$

The easiest way to solve this coupled system is to transform to Fourier space, which yields

(3.88) $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}\displaystyle \unicode[STIX]{x1D714}^{2}-\left(\frac{2p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}+V_{A}^{2}\right)k_{\bot }^{2}+\left(\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}-\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}-V_{A}^{2}\right)k_{\Vert }^{2}; & \quad \displaystyle -\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}k_{\bot }k_{\Vert }\\ \displaystyle -\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}k_{\bot }k_{\Vert }; & \quad \displaystyle \unicode[STIX]{x1D714}^{2}-\frac{3p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}k_{\Vert }^{2}\end{array}\right)\left(\begin{array}{@{}c@{}}u_{x}\\ u_{z}\end{array}\right)=\left(\begin{array}{@{}c@{}}0\\ 0\end{array}\right). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The system has non-trivial solutions only if the determinant of the matrix is zero, yielding the CGL dispersion relation at fourth order in frequency $\unicode[STIX]{x1D714}$ that contains the slow and fast modes

(3.89) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D714}^{4}-\unicode[STIX]{x1D714}^{2}\left[\left(\frac{2p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}+V_{A}^{2}\right)k_{\bot }^{2}+\left(\frac{2p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}+\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}+V_{A}^{2}\right)k_{\Vert }^{2}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,3k_{\Vert }^{2}\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}\left[\left(\frac{2p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}-\frac{{p_{\bot }^{(0)}}^{2}}{3p_{\Vert }^{(0)}\unicode[STIX]{x1D70C}_{0}}+V_{A}^{2}\right)k_{\bot }^{2}-\left(\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}-\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}-V_{A}^{2}\right)k_{\Vert }^{2}\right]=0.\end{eqnarray}$$

One can proceed in several ways with the above dispersion relation.

In the specific cases of strictly parallel ( $k_{\bot }=0$ ) and strictly perpendicular ( $k_{\Vert }=0$ ) propagation directions it is actually easier to work directly with the system (3.88), because the off-diagonal components in the matrix are zero and the dynamics in the $u_{x}$ and $u_{z}$ components decouples. For strictly parallel propagation, one solution is in the $u_{x}$ component, with the dispersion relation identical to the Alfvén wave (3.81). Since the magnetic and velocity fluctuations are only in the $u_{x}$ , $b_{x}$ components, the compressibility $\unicode[STIX]{x1D712}_{b}=0$ and $\unicode[STIX]{x1D712}_{u}=0$ for this mode. The second solution for parallel propagation is in the $u_{z}$ component, and it is called the sound mode, or the ion-acoustic mode, with dispersion relation $\unicode[STIX]{x1D714}=\pm k_{\Vert }\sqrt{3p_{\Vert }^{(0)}/\unicode[STIX]{x1D70C}_{0}}=\pm k_{\Vert }V_{A}\sqrt{3\unicode[STIX]{x1D6FD}_{\Vert }/2}$ . The parallel ion-acoustic mode has $\unicode[STIX]{x1D712}_{u}=1$ and $\unicode[STIX]{x1D712}_{b}=0$ . Considering strictly perpendicular propagation, the waves in the $u_{z}$ component vanish with the solution $\unicode[STIX]{x1D714}=0$ . The waves in the $u_{x}$ component are fast magnetosonic waves with dispersion relation $\unicode[STIX]{x1D714}=\pm k_{\bot }\sqrt{V_{A}^{2}+2p_{\bot }^{(0)}/\unicode[STIX]{x1D70C}_{0}}=\pm k_{\bot }V_{A}\sqrt{1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }}$ , and $\unicode[STIX]{x1D712}_{u}=0$ and $\unicode[STIX]{x1D712}_{b}=1$ . It is also possible to define the parallel and perpendicular sound speeds $C_{\Vert }=\sqrt{3p_{\Vert }^{(0)}/\unicode[STIX]{x1D70C}_{0}}$ ; $C_{\bot }=\sqrt{2p_{\bot }^{(0)}/\unicode[STIX]{x1D70C}_{0}}$ , which yields the parallel acoustic mode dispersion $\unicode[STIX]{x1D714}=\pm k_{\Vert }C_{\Vert }$ and the perpendicular fast mode dispersion $\unicode[STIX]{x1D714}=\pm k_{\bot }\sqrt{V_{A}^{2}+C_{\bot }^{2}}$ .Footnote ⁴

In the general case of oblique propagation, we choose to use the normalized frequencies and wavenumbers (3.29), and rewrite the dispersion relation as

(3.90) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \widetilde{\unicode[STIX]{x1D714}}^{4}-A_{2}\widetilde{\unicode[STIX]{x1D714}}^{2}+A_{0}=0;\\ \displaystyle A_{2}=\widetilde{k}_{\bot }^{2}(1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert })+\widetilde{k}_{\Vert }^{2}\left(1+\unicode[STIX]{x1D6FD}_{\Vert }+{\textstyle \frac{1}{2}}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\right);\\ \displaystyle A_{0}={\textstyle \frac{3}{2}}\widetilde{k}_{\Vert }^{2}\unicode[STIX]{x1D6FD}_{\Vert }\left[\widetilde{k}_{\Vert }^{2}\left(1-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FD}_{\Vert }+{\textstyle \frac{1}{2}}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\right)+\widetilde{k}_{\bot }^{2}\left(1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-{\textstyle \frac{1}{6}}a_{p}^{2}\unicode[STIX]{x1D6FD}_{\Vert }\right)\right].\end{array}\right\} & & \displaystyle\end{eqnarray}$$

It is sometimes useful to introduce the parallel Alfvén phase speed $v_{A\Vert }$ , and its normalization $\widetilde{v}_{A\Vert }\equiv v_{A\Vert }/V_{A}$ , so that

(3.91) $$\begin{eqnarray}v_{A\Vert }=V_{A}\sqrt{1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)};\quad \widetilde{v}_{A\Vert }^{2}=1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1),\end{eqnarray}$$

since $\widetilde{v}_{A\Vert }^{2}$ can be used to write the CGL dispersion relation in a shorter form, as done later in (3.193). The solution of the polynomial (3.90) is simply $\widetilde{\unicode[STIX]{x1D714}}^{2}=(A_{2}\pm \sqrt{A_{2}^{2}-4A_{0}})/2$ , and it is obvious that the coefficient $A_{2}$ is always positive, since both the plasma beta and the temperature anisotropy ratio must always be positive. It is also possible to show that the discriminant under the square root $A_{2}^{2}-4A_{0}\geqslant 0$ (proof shown a few lines below). The solution with the minus sign is called the slow mode and the solution with the plus sign is called the fast mode. For the slow mode, $\widetilde{\unicode[STIX]{x1D714}}^{2}$ becomes negative (and $\widetilde{\unicode[STIX]{x1D714}}$ complex) when $A_{0}<0$ , implying that the slow mode becomes unstable when

(3.92) $$\begin{eqnarray}\left[k_{\Vert }^{2}\left(1-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FD}_{\Vert }+{\textstyle \frac{1}{2}}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\right)+k_{\bot }^{2}\left(1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-{\textstyle \frac{1}{6}}a_{p}^{2}\unicode[STIX]{x1D6FD}_{\Vert }\right)\right]<0,\end{eqnarray}$$

where we suppress the tildes on the wavenumbers since the above condition is valid for both $\widetilde{k}$ and $k$ forms of wavenumbers. The above condition implies that there are two competing instabilities. The highly parallel propagation with $k_{\Vert }\gg k_{\bot }$ yields an instability threshold equivalent to (3.82) and the associated instability is called the parallel firehose instability. The highly oblique propagation with $k_{\bot }\gg k_{\Vert }$ yields the mirror instability and the threshold in the CGL description readsFootnote ⁵

(3.93) $$\begin{eqnarray}1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-\frac{1}{6}a_{p}^{2}\unicode[STIX]{x1D6FD}_{\Vert }<0;\quad \frac{1}{6}\frac{T_{\bot }^{(0)}}{T_{\Vert }^{(0)}}-1-\frac{1}{\unicode[STIX]{x1D6FD}_{\bot }}>0;\quad \frac{1}{6}\frac{\unicode[STIX]{x1D6FD}_{\bot }^{2}}{\unicode[STIX]{x1D6FD}_{\Vert }}>1+\unicode[STIX]{x1D6FD}_{\bot }.\end{eqnarray}$$

It is well documented that the mirror instability threshold in the CGL description is not equivalent to the one found in linear kinetic theory. In particular, the CGL mirror threshold contains a factor of 6 error (Abraham-Shrauner Reference Abraham-Shrauner1967; Tajiri Reference Tajiri1967; Kulsrud Reference Kulsrud, Galeev and Sudan1983; Ferrière & André Reference Ferrière and André2002), since the correct threshold obtained from kinetic theory reads

(3.94) $$\begin{eqnarray}1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-a_{p}^{2}\unicode[STIX]{x1D6FD}_{\Vert }<0;\quad \frac{T_{\bot }^{(0)}}{T_{\Vert }^{(0)}}-1-\frac{1}{\unicode[STIX]{x1D6FD}_{\bot }}>0;\quad \frac{\unicode[STIX]{x1D6FD}_{\bot }^{2}}{\unicode[STIX]{x1D6FD}_{\Vert }}>1+\unicode[STIX]{x1D6FD}_{\bot }.\end{eqnarray}$$

The threshold (3.93) implies that the necessary condition for the mirror instability to exist in the CGL description is $a_{p}=T_{\bot }^{(0)}/T_{\Vert }^{(0)}>6$ , whereas in kinetic theory the necessary condition is $a_{p}>1$ . A very good discussion about the physical mechanism of the mirror instability and the role of resonant particles can be found in Southwood & Kivelson (Reference Southwood and Kivelson1993) and Kivelson & Southwood (Reference Kivelson and Southwood1996). The correct mirror instability threshold is obtained when using the normal closure discussed later on in the text or, more simply, its quasi-static limit (Constantinescu Reference Constantinescu2002; Chust & Belmont Reference Chust and Belmont2006; Passot et al. Reference Passot, Ruban and Sulem2006). In the opposite case where ions are cold, a similar electron temperature anisotropy instability is obtained which is called the field-swelling instability, see e.g. Basu & Coppi (Reference Basu and Coppi1984).

Returning to the oblique dispersion relation, we show that indeed $A_{2}^{2}-4A_{0}\geqslant 0$ . By using (3.78) the wavenumber $k$ can be pulled out of the $A_{0},A_{2}$ coefficients and the dispersion relation can be rewritten in terms of a phase speed as

(3.95) $$\begin{eqnarray}\displaystyle & \displaystyle (\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})^{4}-A_{2}(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})^{2}+A_{0}=0; & \displaystyle\end{eqnarray}$$

(3.96) $$\begin{eqnarray}\displaystyle & \displaystyle A_{2}=1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\left(1-{\textstyle \frac{1}{2}}\cos ^{2}\unicode[STIX]{x1D703}\right)+\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}; & \displaystyle\end{eqnarray}$$

(3.97) $$\begin{eqnarray}\displaystyle & \displaystyle A_{0}={\textstyle \frac{3}{2}}\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}\left[1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\left(1-{\textstyle \frac{1}{2}}\cos ^{2}\unicode[STIX]{x1D703}\right)-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}-{\textstyle \frac{1}{6}}a_{p}^{2}\unicode[STIX]{x1D6FD}_{\Vert }\sin ^{2}\unicode[STIX]{x1D703}\right], & \displaystyle\end{eqnarray}$$

and (after staring for a sufficiently long time) it can be shown that

(3.98) $$\begin{eqnarray}A_{2}^{2}-4A_{0}=\left[1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\left(1-{\textstyle \frac{1}{2}}\cos ^{2}\unicode[STIX]{x1D703}\right)-2\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}\right]^{2}+a_{p}^{2}\unicode[STIX]{x1D6FD}_{\Vert }^{2}\sin ^{2}\unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D703},\end{eqnarray}$$

which is obviously always non-negative. The dispersion relation for the slow and fast waves in the CGL description can be written in the compact form

(3.99) $$\begin{eqnarray}\displaystyle \left(\frac{\widetilde{\unicode[STIX]{x1D714}}}{\widetilde{k}}\right)_{sf}^{2} & = & \displaystyle \left(\frac{\unicode[STIX]{x1D714}}{kV_{A}}\right)_{sf}^{2}=\frac{1}{2}\left[1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\left(1-\frac{1}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)+\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}\right]\nonumber\\ \displaystyle & & \displaystyle \pm \frac{1}{2}\sqrt{\left[1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\left(1-\frac{1}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)-2\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}\right]^{2}+a_{p}^{2}\unicode[STIX]{x1D6FD}_{\Vert }^{2}\sin ^{2}\unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D703}},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

which is equivalent to the dispersion relation of Abraham-Shrauner (Reference Abraham-Shrauner1967), equation (30), here written in the convenient units of $\unicode[STIX]{x1D6FD}_{\Vert }$ and $a_{p}=T_{\bot }^{(0)}/T_{\Vert }^{(0)}$ (for a direct comparison with the units used in that paper, the relations are $\unicode[STIX]{x1D6FD}_{\Vert }=2S_{\Vert }$ and $a_{p}\unicode[STIX]{x1D6FD}_{\Vert }=2S_{\bot }$ ). The magnetic compressibility of the slow and fast modes can be calculated easily directly from the induction equations (3.72). The slow and fast modes generate only components $B_{x}$ and $B_{z}$ with the component $B_{y}=0$ . The transformation of (3.72) to Fourier space yields $-\unicode[STIX]{x1D714}B_{x}=B_{0}k_{\Vert }u_{x}$ , and $-\unicode[STIX]{x1D714}B_{z}=-B_{0}k_{\bot }u_{x}$ , and by applying the magnitude operator yields the magnetic compressibility

(3.100) $$\begin{eqnarray}\unicode[STIX]{x1D712}_{b}(k_{\Vert },k_{\bot })=\frac{k_{\bot }^{2}}{k_{\Vert }^{2}+k_{\bot }^{2}}=\sin ^{2}\unicode[STIX]{x1D703}.\end{eqnarray}$$

For quasi-parallel propagation angles the magnetic compressibility $\unicode[STIX]{x1D712}_{b}$ is therefore not a good indicator for mode recognition, since all three modes have $\unicode[STIX]{x1D712}_{b}$ close to zero. The magnetic compressibility is, however, an excellent tool for oblique propagation angles, because the Alfvén mode has $\unicode[STIX]{x1D712}_{b}=0$ regardless of the propagation direction, whereas for the slow and fast modes $\unicode[STIX]{x1D712}_{b}$ increases with $\unicode[STIX]{x1D703}$ towards $\unicode[STIX]{x1D712}_{b}=1$ . We will leave discussion about the CGL velocity eigenvector (the ratio $\unicode[STIX]{x1D712}_{u}$ ) to a later subsection since the discussion is a bit technical, and we instead focus now on an interesting effect that can be directly obtained by considering the CGL dispersion relations.

Figure 1. Analytic ‘hard’ thresholds of the mirror instability (blue) and the firehose instability (red) at long wavelengths. Solid lines are solutions of the linear kinetic theory (bi-Maxwellian) and crosses are solutions of the linear CGL model. The firehose instability is described correctly (both the oblique firehose and the parallel firehose), whereas the mirror instability contains an asymptotic factor of 6 error for large $\unicode[STIX]{x1D6FD}_{\Vert }$ values. The black dashed line is the CGL threshold (3.103), where for all oblique directions the slow mode speed $v_{s}$ matches the Alfvén mode speed $v_{A}$ , and which separates two regions where $v_{s}<v_{A}$ and $v_{s}>v_{A}$ . The green dotted curve is a special case for parallel propagation (3.108) where $v_{s\Vert }=v_{A\Vert }$ . For parallel propagation, the slow mode speed cannot exceed the Alfvén mode speed, but can only match it, and the green dotted curve separates two regions where $v_{s\Vert }<v_{A\Vert }$ and $v_{s\Vert }=v_{A\Vert }$ . For a similar plot in linear scale, see figure 1 of Abraham-Shrauner (Reference Abraham-Shrauner1967).

3.6 Slow mode can become faster than Alfvén mode

There is a very interesting phenomenon present in the CGL description that is also observed in the kinetic description but that is absent in the MHD description. For a sufficiently high plasma beta, the slow mode phase speed can be larger than the Alfvén mode phase speed. To obtain the critical plasma beta when this happens, one can write the dispersion relation for the Alfvén wave as $(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{A}^{2}=(1-\unicode[STIX]{x1D6FD}_{\Vert }/2+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }/2)\cos ^{2}\unicode[STIX]{x1D703}\equiv A_{A}$ and using this expression with the dispersion relation (3.95), the critical condition when the slow mode speed matches the Alfvén speed is $(A_{2}-\sqrt{A_{2}^{2}-4A_{0}})/2=A_{A}$ , and since $A_{2}^{2}-4A_{0}\geqslant 0$ , this further yields $A_{A}^{2}-A_{2}A_{A}+A_{0}=0$ . It takes quite a few algebraic operations to simplify the result, the easiest way is to collect terms with different powers of $\unicode[STIX]{x1D6FD}_{\Vert }$ and then simplify. A very valuable approach available now is to use mathematical software such as Maple or Mathematica for guidance, and for example the Maple command simplify(expr,trig) immediately simplifies the result, which can then be of course verified with pencil and paper. The result is

(3.101) $$\begin{eqnarray}\displaystyle A_{A}^{2}-A_{2}A_{A}+A_{0}=-\sin ^{2}\,\unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D703}\left[1+\unicode[STIX]{x1D6FD}_{\Vert }\left({\textstyle \frac{3}{2}}a_{p}-2\right)+\unicode[STIX]{x1D6FD}_{\Vert }^{2}a_{p}\left({\textstyle \frac{3}{4}}a_{p}-2\right)\right]=0. & & \displaystyle\end{eqnarray}$$

The expression is naturally satisfied for cases of strictly parallel and strictly perpendicular propagation. For $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ , both the slow mode and the Alfvén mode have zero frequency. For $\unicode[STIX]{x1D703}=0$ , the slow mode speed can be only smaller or equal to the speed of the Alfvén mode (never higher), depending on another condition that will be easier to clarify later when specifically considering the parallel propagation limit. Now, on which side of the threshold (3.101) does the speed of the slow mode become faster than the speed of the Alfvén mode? Working with inequalities is a bit trickier. By assuming that both modes have real frequencies (and are not firehose or mirror unstable), the condition is derived to be

(3.102) $$\begin{eqnarray}\displaystyle -\sin ^{2}\unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D703}\left[1+\unicode[STIX]{x1D6FD}_{\Vert }\left({\textstyle \frac{3}{2}}a_{p}-2\right)+\unicode[STIX]{x1D6FD}_{\Vert }^{2}a_{p}\left({\textstyle \frac{3}{4}}a_{p}-2\right)\right]\geqslant 0, & & \displaystyle\end{eqnarray}$$

which agrees with the result obtained by Abraham-Shrauner (Reference Abraham-Shrauner1967), equation (42). The expression (3.102) reveals that there is a condition that can be satisfied for all directions of propagation and that reads

(3.103) $$\begin{eqnarray}\displaystyle 1+\unicode[STIX]{x1D6FD}_{\Vert }\left({\textstyle \frac{3}{2}}a_{p}-2\right)+\unicode[STIX]{x1D6FD}_{\Vert }^{2}a_{p}\left({\textstyle \frac{3}{4}}a_{p}-2\right)\leqslant 0. & & \displaystyle\end{eqnarray}$$

This is a quadratic equation in $\unicode[STIX]{x1D6FD}_{\Vert }$ which can be solved easily for a given $a_{p}$ (or solved for $a_{p}$ for a given $\unicode[STIX]{x1D6FD}_{\Vert }$ ). To visualize this equation, the threshold is plotted in figure 1 as a black dashed line. A particularly useful result is obtained for isotropic temperatures ( $a_{p}=1$ ). In this case the inequality (3.103) becomes $1-\unicode[STIX]{x1D6FD}_{\Vert }/2-\unicode[STIX]{x1D6FD}_{\Vert }^{2}5/4\leqslant 0$ and the critical plasma beta at which the slow mode speed becomes larger than the Alfvén mode speed is

(3.104) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\Vert }^{\text{crit}}\geqslant \frac{\sqrt{21}-1}{5}\approx 0.72.\end{eqnarray}$$

Thus, this effect exists in the CGL description even when the temperatures are isotropic.

3.7 CGL parallel propagation

When numerically solving the CGL dispersion relations for finite (non-zero) propagation angles, the slow and fast mode expressions always get the correct ordering with $\unicode[STIX]{x1D714}_{s}\leqslant \unicode[STIX]{x1D714}_{f}$ . It is sometimes confusing about how the dispersion relations hold in the limit of parallel propagation $\unicode[STIX]{x1D703}\rightarrow 0$ which we will address here. In this limit, the dispersion relations (3.99) for the slow and fast modes can be directly evaluated as

(3.105) $$\begin{eqnarray}\displaystyle \left(\frac{\widetilde{\unicode[STIX]{x1D714}}}{\widetilde{k}}\right)_{sf}^{2}=\frac{1}{2}\left[1+\frac{1}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }+\unicode[STIX]{x1D6FD}_{\Vert }\right]\pm \frac{1}{2}\sqrt{\left[1+\frac{1}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-2\unicode[STIX]{x1D6FD}_{\Vert }\right]^{2}}. & & \displaystyle\end{eqnarray}$$

Now, because $\sqrt{a^{2}}=|a|$ , for further calculation one needs to determine the sign of the expression under the square root. For $1+\frac{1}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-2\unicode[STIX]{x1D6FD}_{\Vert }>0$ , i.e. when $\unicode[STIX]{x1D6FD}_{\Vert }<2/(4-a_{p})$ , the above equations yield

(3.106) $$\begin{eqnarray}\displaystyle (\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{s}^{2}=\frac{3}{2}\unicode[STIX]{x1D6FD}_{\Vert };\quad (\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{f}^{2}=1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)=(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{A}^{2}, & & \displaystyle\end{eqnarray}$$

so the slow mode is the sound/acoustic mode and the fast mode coincides with the Alfvén mode. However, for $1+\frac{1}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-2\unicode[STIX]{x1D6FD}_{\Vert }<0$ , i.e. when $\unicode[STIX]{x1D6FD}_{\Vert }>2/(4-a_{p})$ , the dispersion relations yield

(3.107) $$\begin{eqnarray}\displaystyle (\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{s}^{2}=1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)=(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{A}^{2};\quad (\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{f}^{2}=\frac{3}{2}\unicode[STIX]{x1D6FD}_{\Vert }, & & \displaystyle\end{eqnarray}$$

so the slow mode coincides with the Alfvén mode and the fast mode is the sound/acoustic mode. In this region, the slow mode phase speed will always stay equal to the Alfvén mode phase speed, regardless of further increases of $\unicode[STIX]{x1D6FD}_{\Vert }$ . In the special case when $1+\frac{1}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-2\unicode[STIX]{x1D6FD}_{\Vert }=0$ , which is equivalent to

(3.108) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\Vert }=\frac{2}{4-a_{p}};\quad a_{p}=4-\frac{2}{\unicode[STIX]{x1D6FD}_{\Vert }},\end{eqnarray}$$

all three modes propagate with the same phase speed $(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{Asf}^{2}=\frac{3}{2}\unicode[STIX]{x1D6FD}_{\Vert }$ . The criterion (3.108) can be satisfied only for $\unicode[STIX]{x1D6FD}_{\Vert }\geqslant 1/2$ and in the limit $\unicode[STIX]{x1D6FD}_{\Vert }\rightarrow \infty$ the asymptote is $a_{p}=4$ , so the possible range of $a_{p}$ is $[0,4]$ . This curve is plotted in figure 1 as a green dotted line. The curve separates two regions. In the first region $\unicode[STIX]{x1D6FD}_{\Vert }<2/(4-a_{p})$ and $v_{s\Vert }<v_{A\Vert }$ . In the second region $\unicode[STIX]{x1D6FD}_{\Vert }>2/(4-a_{p})$ and $v_{s\Vert }=v_{A\Vert }$ . The interesting result to remember is that, for isotropic temperatures ( $a_{p}=1$ ) and parallel propagation, the critical plasma beta at which the slow mode speed matches the Alfvén mode speed is

(3.109) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\Vert }^{\text{crit}}(0)\geqslant {\textstyle \frac{2}{3}}.\end{eqnarray}$$

3.8 CGL with $T_{\bot }=T_{\Vert }$ is not equivalent to MHD

It is important to emphasize that, even for isotropic temperatures $a_{p}=1$ , the CGL dispersion (3.99) is not equivalent to the MHD dispersion for the slow and fast waves. The MHD dispersion is usually written in the form (see (C 20) in appendix C)

(3.110) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x1D714}}{k}\right)^{2}=\frac{1}{2}(V_{A}^{2}+C_{s}^{2})\pm \frac{1}{2}\sqrt{(V_{A}^{2}+C_{s}^{2})^{2}-4V_{A}^{2}C_{s}^{2}\cos ^{2}\unicode[STIX]{x1D703}}, & & \displaystyle\end{eqnarray}$$

or alternatively

(3.111) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x1D714}}{k}\right)^{2}=\frac{1}{2}(V_{A}^{2}+C_{s}^{2})\pm \frac{1}{2}\sqrt{(V_{A}^{2}-C_{s}^{2})^{2}+4V_{A}^{2}C_{s}^{2}\sin ^{2}\unicode[STIX]{x1D703}}, & & \displaystyle\end{eqnarray}$$

which nicely shows that the expression under the square root is always greater or equal to zero. The MHD sound speed is defined as $C_{s}^{2}=\unicode[STIX]{x1D6FE}p^{(0)}/\unicode[STIX]{x1D70C}_{0}$ , where $\unicode[STIX]{x1D6FE}=5/3$ . Rewritten with the parameters we use here $C_{s}^{2}/V_{A}^{2}=\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}_{\Vert }/2=5\unicode[STIX]{x1D6FD}_{\Vert }/6$ , the MHD dispersion for the slow and fast waves reads

(3.112) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x1D714}}{kV_{A}}\right)_{\text{MHD}}^{2}=\frac{1}{2}\left(1+\frac{\unicode[STIX]{x1D6FE}}{2}\unicode[STIX]{x1D6FD}_{\Vert }\right)\pm \frac{1}{2}\sqrt{\left(1+\frac{\unicode[STIX]{x1D6FE}}{2}\unicode[STIX]{x1D6FD}_{\Vert }\right)^{2}-2\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}}, & & \displaystyle\end{eqnarray}$$

or alternatively

(3.113) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x1D714}}{kV_{A}}\right)_{\text{MHD}}^{2}=\frac{1}{2}\left(1+\frac{\unicode[STIX]{x1D6FE}}{2}\unicode[STIX]{x1D6FD}_{\Vert }\right)\pm \frac{1}{2}\sqrt{\left(1-\frac{\unicode[STIX]{x1D6FE}}{2}\unicode[STIX]{x1D6FD}_{\Vert }\right)^{2}+2\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}_{\Vert }\sin ^{2}\unicode[STIX]{x1D703}}. & & \displaystyle\end{eqnarray}$$

In the CGL description with isotropic temperatures $(a_{p}=1)$ the dispersion relation (3.99) for slow and fast waves reads

(3.114) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x1D714}}{kV_{A}}\right)_{\text{CGL}}^{2} & = & \displaystyle \frac{1}{2}\left[1+\unicode[STIX]{x1D6FD}_{\Vert }\left(1+\frac{1}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)\right]\nonumber\\ \displaystyle & & \displaystyle \pm \,\frac{1}{2}\sqrt{\left[1+\unicode[STIX]{x1D6FD}_{\Vert }\left(1-\frac{5}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)\right]^{2}+\unicode[STIX]{x1D6FD}_{\Vert }^{2}\sin ^{2}\unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D703}},\end{eqnarray}$$

or alternatively

(3.115) $$\begin{eqnarray}\displaystyle \left(\frac{\unicode[STIX]{x1D714}}{kV_{A}}\right)_{\text{CGL}}^{2} & = & \displaystyle \frac{1}{2}\left[1+\unicode[STIX]{x1D6FD}_{\Vert }\left(1+\frac{1}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)\right]\nonumber\\ \displaystyle & & \displaystyle \pm \,\frac{1}{2}\sqrt{\left[1+\unicode[STIX]{x1D6FD}_{\Vert }\left(1+\frac{1}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)\right]^{2}-6\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}\left(1+\frac{5}{6}\unicode[STIX]{x1D6FD}_{\Vert }\sin ^{2}\unicode[STIX]{x1D703}\right)}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

We wrote down all major possibilities on purpose, since we wanted to clearly show that, regardless of the choice of $\unicode[STIX]{x1D6FD}_{\Vert }$ and $\unicode[STIX]{x1D6FE}$ , the MHD dispersion relation cannot match the CGL dispersion relation simultaneously for all propagation directions $\unicode[STIX]{x1D703}$ . One could even play with an abstract idea to make the MHD $\unicode[STIX]{x1D6FE}$ dependent on the angle $\unicode[STIX]{x1D703}$ , and possibly $\unicode[STIX]{x1D6FD}_{\Vert }$ . Even then, there is no obvious way to modify the MHD dispersion relation so that it behaves like the CGL model. Additionally, the CGL description leads to the development of pressure anisotropy even when the initial pressure is isotropic.

3.9 MHD velocity eigenvector

Before we proceed with the discussion of the CGL velocity eigenvector (which can appear to be a bit tricky), for the sake of clarity it is useful to consider the simpler MHD case first. By following the steps outlined above, it is easy to show that again the Alfvén mode dynamics in the $u_{y}$ decouples from the dynamics in the other directions and the velocity eigenvector matrix for the slow and fast waves in MHD is

(3.116) $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D714}^{2}-(V_{A}^{2}k^{2}+C_{s}^{2}k_{\bot }^{2}) & -C_{s}^{2}k_{\Vert }k_{\bot }\\ -C_{s}^{2}k_{\Vert }k_{\bot } & \unicode[STIX]{x1D714}^{2}-C_{s}^{2}k_{\Vert }^{2}\end{array}\right)\left(\begin{array}{@{}c@{}}u_{x}\\ u_{z}\end{array}\right)=\left(\begin{array}{@{}c@{}}0\\ 0\end{array}\right), & & \displaystyle\end{eqnarray}$$

or alternatively

(3.117) $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}(\unicode[STIX]{x1D714}/k)^{2}-(V_{A}^{2}+C_{s}^{2}\sin ^{2}\unicode[STIX]{x1D703}) & -C_{s}^{2}\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}\\ -C_{s}^{2}\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703} & (\unicode[STIX]{x1D714}/k)^{2}-C_{s}^{2}\cos ^{2}\unicode[STIX]{x1D703}\end{array}\right)\left(\begin{array}{@{}c@{}}u_{x}\\ u_{z}\end{array}\right)=\left(\begin{array}{@{}c@{}}0\\ 0\end{array}\right). & & \displaystyle\end{eqnarray}$$

This yields the MHD dispersion relations for the slow and fast modes in the form of (3.110) or alternatively (3.111). It possible to either express the ratio $u_{x}/u_{z}$ or $u_{z}/u_{x}$ . We want to calculate the velocity ratio $\unicode[STIX]{x1D712}_{u}$ and we choose the first possibility that allows us to express that for the slow and fast modes (since $u_{y}=0$ )

(3.118) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{u}(k_{\Vert },k_{\bot })=\frac{|u_{z}|^{2}}{|u_{x}|^{2}+|u_{z}|^{2}}=\frac{1}{\displaystyle \left|\frac{u_{x}}{u_{z}}\right|^{2}+1}. & & \displaystyle\end{eqnarray}$$

The ratio $u_{x}/u_{z}$ can be found for example from the second row of the matrix as

(3.119) $$\begin{eqnarray}\frac{u_{x}}{u_{z}}=\frac{(\unicode[STIX]{x1D714}/k)^{2}-C_{s}^{2}\cos ^{2}\unicode[STIX]{x1D703}}{C_{s}^{2}\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}},\end{eqnarray}$$

or equivalently from the first row of the matrix as

(3.120) $$\begin{eqnarray}\frac{u_{x}}{u_{z}}=\frac{C_{s}^{2}\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}}{(\unicode[STIX]{x1D714}/k)^{2}-(V_{A}^{2}+C_{s}^{2}\sin ^{2}\unicode[STIX]{x1D703})}.\end{eqnarray}$$

The possible range of the ratio $u_{x}/u_{z}$ is from $-\infty$ to $+\infty$ and the possible range for $\unicode[STIX]{x1D712}_{u}$ is from $0$ to $1$ . Solutions can be easily divided into two categories – depending on the behaviour for parallel propagation – and are plotted in figure 2. For the sake of clarity, we only consider propagation with positive wavenumbers $k_{\Vert }$ and $k_{\bot }$ , so the possible range of angle $\unicode[STIX]{x1D703}$ is from $0$ to $\unicode[STIX]{x03C0}/2$ and $\sin \unicode[STIX]{x1D703}\geqslant 0$ , $\cos \unicode[STIX]{x1D703}\geqslant 0$ . Consider first the case $C_{s}<V_{A}$ . For parallel propagation, the dispersion relation (3.111) directly yields that for the fast mode $(\unicode[STIX]{x1D714}/k)_{f}^{2}|_{\unicode[STIX]{x1D703}=0}=V_{A}^{2}$ and for the slow mode $(\unicode[STIX]{x1D714}/k)_{s}^{2}|_{\unicode[STIX]{x1D703}=0}=C_{s}^{2}$ . Using the expression (3.119) then yields that for the fast mode $(u_{x}/u_{z})_{f}|_{\unicode[STIX]{x1D703}=0}=+\infty$ and for the slow mode $(u_{x}/u_{z})_{s}|_{\unicode[STIX]{x1D703}=0}=0$ . This implies that for $C_{s}<V_{A}$ , all the $\unicode[STIX]{x1D712}_{u}(\unicode[STIX]{x1D703})$ curves start at

(3.121) $$\begin{eqnarray}C_{s}<V_{A}:\quad \unicode[STIX]{x1D712}_{u}^{f}(\unicode[STIX]{x1D703}=0)=0;\quad \unicode[STIX]{x1D712}_{u}^{s}(\unicode[STIX]{x1D703}=0)=1,\end{eqnarray}$$

regardless of the actual value of $C_{s}/V_{A}<1$ . The second case $C_{s}>V_{A}$ is straightforward, since for parallel propagation everything is just turned around; the fast mode dispersion relation is $(\unicode[STIX]{x1D714}/k)_{f}^{2}|_{\unicode[STIX]{x1D703}=0}=C_{s}^{2}$ and the slow mode dispersion relation is $(\unicode[STIX]{x1D714}/k)_{s}^{2}|_{\unicode[STIX]{x1D703}=0}=V_{A}^{2}$ . Implying that for $C_{s}>V_{A}$ , all the $\unicode[STIX]{x1D712}_{u}(\unicode[STIX]{x1D703})$ curves start at

(3.122) $$\begin{eqnarray}C_{s}>V_{A}:\quad \unicode[STIX]{x1D712}_{u}^{f}(\unicode[STIX]{x1D703}=0)=1;\quad \unicode[STIX]{x1D712}_{u}^{s}(\unicode[STIX]{x1D703}=0)=0,\end{eqnarray}$$

regardless of the actual value of $C_{s}/V_{A}>1$ . The perpendicular propagation limit is even easier since the split into two categories is not required and for both cases the dispersion (3.110) yields $(\unicode[STIX]{x1D714}/k)_{f}^{2}|_{\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2}=V_{A}^{2}+C_{s}^{2}$ and $(\unicode[STIX]{x1D714}/k)_{s}^{2}|_{\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2}=0$ , implying that for the fast mode $(u_{x}/u_{z})_{f}|_{\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2}=+\infty$ and that for the slow mode $(u_{x}/u_{z})_{s}|_{\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2}=0$ . All the $\unicode[STIX]{x1D712}_{u}(\unicode[STIX]{x1D703})$ curves therefore ‘end’ at

(3.123) $$\begin{eqnarray}\unicode[STIX]{x1D712}_{u}^{f}(\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2)=0;\quad \unicode[STIX]{x1D712}_{u}^{s}(\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2)=1,\end{eqnarray}$$

whether the ratio $C_{s}/V_{A}<1$ or $C_{s}/V_{A}>1$ . The behaviour of the $\unicode[STIX]{x1D712}_{u}(\unicode[STIX]{x1D703})$ curves is plotted in figure 2.

In MHD, the ratio $C_{s}^{2}/V_{A}^{2}$ is sometimes used as the definition of plasma beta, which we will call here $\unicode[STIX]{x1D6FD}_{\text{MHD}}\equiv C_{s}^{2}/V_{A}^{2}$ . As can be seen in figure 2, for $\unicode[STIX]{x1D6FD}_{\text{MHD}}$ approaching 1, the curves display a ‘singular’ behaviour, and it is difficult to guess what will happen when $\unicode[STIX]{x1D6FD}_{\text{MHD}}$ reaches the value of 1 exactly. This special case is addressed in the next subsection. There are also two special cases, one for $C_{s}\ll V_{A}$ or $\unicode[STIX]{x1D6FD}_{\text{MHD}}\ll 1$ and one for $C_{s}\gg V_{A}$ or $\unicode[STIX]{x1D6FD}_{\text{MHD}}\gg 1$ . Both limits $C_{s}\ll V_{A}$ and $C_{s}\gg V_{A}$ are easily obtained by rewriting the dispersion relation (3.110) to a form

(3.124) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x1D714}}{k}\right)^{2}=\frac{1}{2}(V_{A}^{2}+C_{s}^{2})\pm \frac{1}{2}(V_{A}^{2}+C_{s}^{2})\sqrt{1-\frac{4V_{A}^{2}C_{s}^{2}}{(V_{A}^{2}+C_{s}^{2})^{2}}\cos ^{2}\unicode[STIX]{x1D703}}.\end{eqnarray}$$

In both limits the second term under the square root is small and the expression can be expanded as $\sqrt{1-\unicode[STIX]{x1D716}}=1-\unicode[STIX]{x1D716}/2$ , yielding the slow and fast mode dispersion relations

(3.125) $$\begin{eqnarray}C_{s}\ll V_{A}~\text{or}~C_{s}\gg V_{A}:\quad \left(\frac{\unicode[STIX]{x1D714}}{k}\right)_{f}^{2}=V_{A}^{2}+C_{s}^{2}-\frac{V_{A}^{2}C_{s}^{2}}{V_{A}^{2}+C_{s}^{2}}\cos ^{2}\unicode[STIX]{x1D703};\quad \left(\frac{\unicode[STIX]{x1D714}}{k}\right)_{s}^{2}=\frac{V_{A}^{2}C_{s}^{2}}{V_{A}^{2}+C_{s}^{2}}\cos ^{2}\unicode[STIX]{x1D703},\end{eqnarray}$$

which further yield dispersion relations (3.126), (3.128). Note that the limit $C_{s}\ll V_{A}$ can be easily obtained directly from the velocity matrix (3.117) by neglecting the off-diagonal terms that decouple the two modes. In contrast, the limit $C_{s}\gg V_{A}$ is not that obvious and one needs to calculate the determinant.

In the limit $\unicode[STIX]{x1D6FD}_{\text{MHD}}\ll 1$ , the MHD dispersion relation simplifies to

(3.126) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\text{MHD}}\ll 1:\quad (\unicode[STIX]{x1D714}/k)_{f}^{2}=V_{A}^{2}+C_{s}^{2}\sin ^{2}\unicode[STIX]{x1D703};\quad (\unicode[STIX]{x1D714}/k)_{s}^{2}=C_{s}^{2}\cos ^{2}\unicode[STIX]{x1D703}.\end{eqnarray}$$

The expression (3.119) yields for the fast mode $(u_{x}/u_{z})_{f}(\unicode[STIX]{x1D703})=\infty$ and the expression (3.120) yields for the slow mode $(u_{x}/u_{z})_{s}(\unicode[STIX]{x1D703})=0$ , implying

(3.127) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\text{MHD}}\ll 1:\quad \unicode[STIX]{x1D712}_{u}^{f}(\unicode[STIX]{x1D703})=0;\quad \unicode[STIX]{x1D712}_{u}^{s}(\unicode[STIX]{x1D703})=1.\end{eqnarray}$$

In the opposite limit $\unicode[STIX]{x1D6FD}_{\text{MHD}}\gg 1$ , the MHD dispersions simplify to

(3.128) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\text{MHD}}\gg 1:\quad (\unicode[STIX]{x1D714}/k)_{f}^{2}=C_{s}^{2}+V_{A}^{2}\sin ^{2}\unicode[STIX]{x1D703};\quad (\unicode[STIX]{x1D714}/k)_{s}^{2}=V_{A}^{2}\cos ^{2}\unicode[STIX]{x1D703}.\end{eqnarray}$$

In this limit, the expression (3.119) yields $(u_{x}/u_{z})_{f}(\unicode[STIX]{x1D703})=(1+V_{A}^{2}/C_{s}^{2})\tan \unicode[STIX]{x1D703}$ , that describes the behaviour of the fast mode for large beta values and the complete limit is $(u_{x}/u_{z})_{f}(\unicode[STIX]{x1D703})=\tan \unicode[STIX]{x1D703}$ . For the slow mode the expression (3.119) yields $(u_{x}/u_{z})_{s}(\unicode[STIX]{x1D703})=-(1-V_{A}^{2}/C_{s}^{2})\cot \unicode[STIX]{x1D703}$ and the complete limit is $(u_{x}/u_{z})_{s}(\unicode[STIX]{x1D703})=-\cot \unicode[STIX]{x1D703}$ . The $\unicode[STIX]{x1D712}_{u}(\unicode[STIX]{x1D703})$ solutions for large beta values therefore converge to

(3.129) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\text{MHD}}\gg 1:\quad \unicode[STIX]{x1D712}_{u}^{f}(\unicode[STIX]{x1D703})=\cos ^{2}\unicode[STIX]{x1D703};\quad \unicode[STIX]{x1D712}_{u}^{s}(\unicode[STIX]{x1D703})=\sin ^{2}\unicode[STIX]{x1D703},\end{eqnarray}$$

and these two curves are plotted in figure 2 as dashed lines.

Figure 2. MHD velocity ratio $\unicode[STIX]{x1D712}_{u}(\unicode[STIX]{x1D703})=|u_{\Vert }|^{2}/|u|^{2}$ for the slow mode (red lines) and the fast mode (blue lines) for different ratios of $C_{s}^{2}/V_{A}^{2}=\unicode[STIX]{x1D6FD}_{\text{MHD}}$ . Panel (a) is for $\unicode[STIX]{x1D6FD}_{\text{MHD}}<1$ , and the thickness of the lines increases with $\unicode[STIX]{x1D6FD}_{\text{MHD}}$ when approaching the critical value of $\unicode[STIX]{x1D6FD}_{\text{MHD}}=1$ . The plotted lines have $\unicode[STIX]{x1D6FD}_{\text{MHD}}=0.3;0.6;0.8;0.9;0.99$ . Panel (b) is for $\unicode[STIX]{x1D6FD}_{\text{MHD}}>1$ and the thickness of the lines decreases with $\unicode[STIX]{x1D6FD}_{\text{MHD}}$ when going away from the critical value of $\unicode[STIX]{x1D6FD}_{\text{MHD}}=1$ . The plotted lines have $\unicode[STIX]{x1D6FD}_{\text{MHD}}=1.01;1.1;1.2;2.0;10;10^{3}$ . In both figures, the dotted lines are the critical case $\unicode[STIX]{x1D6FD}_{\text{MHD}}=1$ with solutions (3.134). In (b) the dashed lines represent the limit $\unicode[STIX]{x1D6FD}_{\text{MHD}}\gg 1$ with solutions (3.129) and the dashed lines nicely overlap with a thin solid line solutions obtained for $\unicode[STIX]{x1D6FD}_{\text{MHD}}=10^{3}$ . The Alfvén mode has $\unicode[STIX]{x1D712}_{u}(\unicode[STIX]{x1D703})=0$ regardless of the plasma beta and is not plotted.

3.9.1 MHD special case $V_{A}=C_{s}$

Consider the very peculiar case of parallel propagation $\unicode[STIX]{x1D703}=0$ with $V_{A}=C_{s}$ . The velocity eigenvector matrix (3.117), when evaluated exactly at these values reads

(3.130) $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}0 & 0\\ 0 & 0\end{array}\right)\left(\begin{array}{@{}c@{}}u_{x}\\ u_{z}\end{array}\right)=\left(\begin{array}{@{}c@{}}0\\ 0\end{array}\right). & & \displaystyle\end{eqnarray}$$

It is not possible to determine the ratio $u_{x}/u_{z}$ and the eigenvector must be found by taking the limit $\unicode[STIX]{x1D703}\rightarrow 0$ . For $V_{A}=C_{s}$ the dispersion relation for the slow and fast modes (3.111) simplifies to $(\unicode[STIX]{x1D714}/k)^{2}=V_{A}^{2}\pm V_{A}^{2}|\sin \unicode[STIX]{x1D703}|$ . As before, for the sake of simplicity we consider only positive wavenumbers, so the propagation angle $\unicode[STIX]{x1D703}$ is in the range $[0,\unicode[STIX]{x03C0}/2]$ and $\sin \unicode[STIX]{x1D703}\geqslant 0$ , $\cos \unicode[STIX]{x1D703}\geqslant 0$ . Evaluating the velocity matrix (3.117) at $V_{A}=C_{s}$ only, it is possible to pull the $\sin \unicode[STIX]{x1D703}$ out of the system so that

(3.131) $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}\pm 1-\sin \unicode[STIX]{x1D703} & -\cos \unicode[STIX]{x1D703}\\ -\cos \unicode[STIX]{x1D703} & \pm 1+\sin \unicode[STIX]{x1D703}\end{array}\right)V_{A}^{2}\sin \unicode[STIX]{x1D703}\left(\begin{array}{@{}c@{}}u_{x}\\ u_{z}\end{array}\right)=\left(\begin{array}{@{}c@{}}0\\ 0\end{array}\right), & & \displaystyle\end{eqnarray}$$

which nicely demonstrates that for $\unicode[STIX]{x1D703}\rightarrow 0$ the system approaches (3.130). Nevertheless, we can now calculate the velocity ratio (for example from the second row) as

(3.132) $$\begin{eqnarray}\frac{u_{x}}{u_{z}}=\frac{\pm 1+\sin \unicode[STIX]{x1D703}}{\cos \unicode[STIX]{x1D703}},\end{eqnarray}$$

which can be evaluated at $\unicode[STIX]{x1D703}=0$ . For the fast mode $(u_{x}/u_{z})^{f}(0)=1$ and for the slow mode $(u_{x}/u_{z})^{s}(0)=-1$ . This implies that for the peculiar case of $V_{A}=C_{s}$ , the velocity ratio $\unicode[STIX]{x1D712}_{u}^{sf}(0)=1/2$ for both the fast and slow modes. The expression (3.132) can of course be directly obtained by using the result $(\unicode[STIX]{x1D714}/k)^{2}=V_{A}^{2}\pm V_{A}^{2}\sin \unicode[STIX]{x1D703}$ in the expression (3.119) and by cancelling the $\sin \unicode[STIX]{x1D703}$ in the nominator and denominator. The expression (3.132) is valid for any angle $\unicode[STIX]{x1D703}$ . However, for the slow mode it becomes singular at $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ , and has to be rearranged by multiplying and dividing with $1+\sin \unicode[STIX]{x1D703}$ (or equivalently, use the first row of the matrix). To have useful expression for the entire range of $\unicode[STIX]{x1D703}$ , we therefore split the expressions for the slow and fast modes as

(3.133) $$\begin{eqnarray}(u_{x}/u_{z})^{s}=-\frac{\cos \unicode[STIX]{x1D703}}{1+\sin \unicode[STIX]{x1D703}};\quad (u_{x}/u_{z})^{f}=\frac{1+\sin \unicode[STIX]{x1D703}}{\cos \unicode[STIX]{x1D703}},\end{eqnarray}$$

which yields, in this peculiar case of $V_{A}=C_{s}$ , that the ratio $\unicode[STIX]{x1D712}_{u}$ for the slow and fast modes can be expressed as

(3.134) $$\begin{eqnarray}V_{A}=C_{s}:\quad \unicode[STIX]{x1D712}_{u}^{s}(\unicode[STIX]{x1D703})={\textstyle \frac{1}{2}}(1+\sin \unicode[STIX]{x1D703});\quad \unicode[STIX]{x1D712}_{u}^{f}(\unicode[STIX]{x1D703})={\textstyle \frac{1}{2}}(1-\sin \unicode[STIX]{x1D703}).\end{eqnarray}$$

These solutions are plotted in figure 2 as dotted lines.

3.10 CGL velocity eigenvector

To find the ratio for velocity components $\unicode[STIX]{x1D712}_{u}$ in the CGL description, it is useful for a moment to work with abbreviated symbols, and we write the matrix (3.88) for the $u_{x}$ and $u_{z}$ components as

(3.135) $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})^{2}-a; & -b\\ -b & (\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})^{2}-c\end{array}\right)\left(\begin{array}{@{}c@{}}u_{x}\\ u_{z}\end{array}\right)=\left(\begin{array}{@{}c@{}}0\\ 0\end{array}\right), & & \displaystyle\end{eqnarray}$$

where the coefficients are

(3.136) $$\begin{eqnarray}\displaystyle a=1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\left(1-\frac{1}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)-\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}\cos ^{2}\unicode[STIX]{x1D703};\quad b=\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}a_{p}\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703};\quad c=\frac{3}{2}\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The determinant must be zero, which implies $(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})^{4}-(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})^{2}(a+c)+ac-b^{2}=0$ , and has a solution

(3.137) $$\begin{eqnarray}(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})^{2}={\textstyle \frac{1}{2}}(a+c)\pm {\textstyle \frac{1}{2}}\sqrt{(a+c)^{2}-4ac+4b^{2}}={\textstyle \frac{1}{2}}(a+c)\pm {\textstyle \frac{1}{2}}\sqrt{(a-c)^{2}+4b^{2}}.\end{eqnarray}$$

The last step was actually used in obtaining (3.98) which is otherwise quite difficult to see. Using this result in the above matrix, the eigenvector matrix becomes

(3.138) $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}-\frac{1}{2}(a-c)\pm \frac{1}{2}\sqrt{(a-c)^{2}+4b^{2}}; & -b\\ -b & \frac{1}{2}(a-c)\pm \frac{1}{2}\sqrt{(a-c)^{2}+4b^{2}}\end{array}\right)\left(\begin{array}{@{}c@{}}u_{x}\\ u_{z}\end{array}\right)=\left(\begin{array}{@{}c@{}}0\\ 0\end{array}\right), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where

(3.139) $$\begin{eqnarray}\displaystyle a-c=1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\left(1-{\textstyle \frac{1}{2}}\cos ^{2}\unicode[STIX]{x1D703}\right)-2\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}. & & \displaystyle\end{eqnarray}$$

The ratio of $u_{x}/u_{z}$ can be expressed (for example from the second row) as

(3.140) $$\begin{eqnarray}\displaystyle \frac{u_{x}}{u_{z}} & = & \displaystyle \frac{1}{b}[(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})^{2}-c]=\frac{1}{2b}[a-c\pm \sqrt{(a-c)^{2}+4b^{2}}]\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{\unicode[STIX]{x1D6FD}_{\Vert }a_{p}\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}}\left[1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\left(1-\frac{1}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)-2\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}\right.\nonumber\\ \displaystyle & & \displaystyle \pm \left.\sqrt{\left(1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\left(1-\frac{1}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)-2\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}\right)^{2}+a_{p}^{2}\unicode[STIX]{x1D6FD}_{\Vert }^{2}\sin ^{2}\unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D703}}\right],\end{eqnarray}$$

or alternatively (from the first row) as

(3.141) $$\begin{eqnarray}\displaystyle \frac{u_{x}}{u_{z}} & = & \displaystyle \frac{b}{(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})^{2}-a}=\frac{2b}{-(a-c)\pm \sqrt{(a-c)^{2}+4b^{2}}}\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D6FD}_{\Vert }a_{p}\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}\left[-1-a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\left(1-\frac{1}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)+2\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}\right.\nonumber\\ \displaystyle & & \displaystyle \pm \left.\sqrt{\left(1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\left(1-\frac{1}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)-2\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}\right)^{2}+a_{p}^{2}\unicode[STIX]{x1D6FD}_{\Vert }^{2}\sin ^{2}\unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D703}}\right]^{-1}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The behaviour of these solutions is plotted in figure 3, where we plot solutions for isotropic temperatures $a_{p}=1$ . Similarly to MHD, two classes of solutions can be characterized based upon the behaviour for parallel propagation $\unicode[STIX]{x1D703}=0$ , depending if the quantity $a-c$ (3.139) evaluated at $\unicode[STIX]{x1D703}=0$ is positive or negative. The quantity $(a-c)|_{\unicode[STIX]{x1D703}=0}=1+\frac{1}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-2\unicode[STIX]{x1D6FD}_{\Vert }$ was discussed in the section for parallel propagation and this quantity represents the green dotted curve in figure 1 that separates the system into two regions. For $(a-c)|_{\unicode[STIX]{x1D703}=0}>0$ , i.e. in the region $\unicode[STIX]{x1D6FD}_{\Vert }<2/(4-a_{p})$ , for the slow mode (use (3.141)) the ratio $(u_{x}/u_{z})^{s}(0)=0$ and for the fast mode (use (3.140)) the ratio $(u_{x}/u_{z})^{f}(0)=+\infty$ . In this region, all the $\unicode[STIX]{x1D712}_{u}(\unicode[STIX]{x1D703})$ curves start at

(3.142) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\Vert }<2/(4-a_{p}):\quad \unicode[STIX]{x1D712}_{u}^{s}(0)=1;\quad \unicode[STIX]{x1D712}_{u}^{f}(0)=0.\end{eqnarray}$$

For $(a-c)|_{\unicode[STIX]{x1D703}=0}<0$ , i.e. in the region $\unicode[STIX]{x1D6FD}_{\Vert }>2/(4-a_{p})$ , for the slow mode (use (3.140)) the ratio $(u_{x}/u_{z})^{s}(0)=+\infty$ and for the fast mode (use (3.141)) the ratio $(u_{x}/u_{z})^{f}(0)=0$ . In this region, all the $\unicode[STIX]{x1D712}_{u}(\unicode[STIX]{x1D703})$ curves start at

(3.143) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\Vert }>2/(4-a_{p}):\quad \unicode[STIX]{x1D712}_{u}^{s}(0)=0;\quad \unicode[STIX]{x1D712}_{u}^{f}(0)=1.\end{eqnarray}$$

For perpendicular propagation $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ the quantity $(a-c)|_{\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2}=1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }$ is always positive and the separation into two regions is not necessary. For the slow mode $(u_{x}/u_{z})^{s}(\unicode[STIX]{x03C0}/2)=0$ and for the fast mode $(u_{x}/u_{z})^{f}(\unicode[STIX]{x03C0}/2)=+\infty$ . All the $\unicode[STIX]{x1D712}_{u}(\unicode[STIX]{x1D703})$ curves, regardless of the value of $\unicode[STIX]{x1D6FD}_{\Vert }$ therefore end at

(3.144) $$\begin{eqnarray}\quad \unicode[STIX]{x1D712}_{u}^{s}(\unicode[STIX]{x03C0}/2)=1;\quad \unicode[STIX]{x1D712}_{u}^{f}(\unicode[STIX]{x03C0}/2)=0.\end{eqnarray}$$

This summarizes the general behaviour of the solutions. However, as can be seen from figure 3, there is a special case when the parameters approach $\unicode[STIX]{x1D6FD}_{\Vert }=2/(4-a_{p})$ , i.e. when $(a-c)|_{\unicode[STIX]{x1D703}=0}=0$ , that will be discussed in the next subsection. One can again consider the two limiting cases when $\unicode[STIX]{x1D6FD}_{\Vert }$ is small or large. The $\unicode[STIX]{x1D6FD}_{\Vert }\ll 1$ case yields dispersion relationsFootnote ⁶

(3.146) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{\Vert }\ll 1:\quad (\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{f}^{2} & = & \displaystyle 1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\left(1-\frac{1}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)-\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}\cos ^{2}\unicode[STIX]{x1D703};\end{eqnarray}$$

(3.147) $$\begin{eqnarray}\displaystyle (\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{s}^{2} & = & \displaystyle {\textstyle \frac{3}{2}}\unicode[STIX]{x1D6FD}_{\Vert }\cos ^{2}\unicode[STIX]{x1D703}.\end{eqnarray}$$

Using expression (3.141) for the slow mode yields $(u_{x}/u_{z})^{s}(\unicode[STIX]{x1D703})=0$ and expression (3.140) for the fast mode yields $(u_{x}/u_{z})^{f}(\unicode[STIX]{x1D703})=\infty$ , implying that, in low beta limit, the $\unicode[STIX]{x1D712}_{u}(\unicode[STIX]{x1D703})$ solutions are

(3.148) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\Vert }\ll 1:\quad \unicode[STIX]{x1D712}_{u}^{s}(\unicode[STIX]{x1D703})=1;\quad \unicode[STIX]{x1D712}_{u}^{f}(\unicode[STIX]{x1D703})=0,\end{eqnarray}$$

and the constant solutions approaching this limit are visible in the plot of figure 3. The opposite $\unicode[STIX]{x1D6FD}_{\Vert }\gg 1$ limit in the CGL description is not particularly revealing since there is no obvious way to simplify the expansion, even for isotropic temperatures. For $a_{p}=1$ , the $\unicode[STIX]{x1D6FD}_{\Vert }\gg 1$ expansion yields

(3.149) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{\Vert }\gg 1: & & \displaystyle (\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{sf}^{2}=\frac{1}{2}\left(1+\unicode[STIX]{x1D6FD}_{\Vert }\left(1+\frac{1}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)\right)\pm \frac{1}{2}\left[\unicode[STIX]{x1D6FD}_{\Vert }\sqrt{A}+\frac{1-\frac{5}{2}\cos ^{2}\unicode[STIX]{x1D703}}{\sqrt{A}}\right],\end{eqnarray}$$

(3.150) $$\begin{eqnarray}\displaystyle & & \displaystyle A=\left(1-{\textstyle \frac{5}{2}}\cos ^{2}\unicode[STIX]{x1D703}\right)^{2}+\sin ^{2}\unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D703},\end{eqnarray}$$

where we neglected a term proportional to $1/\unicode[STIX]{x1D6FD}_{\Vert }$ . The expression results in the correct parallel limit $(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{f}^{2}(0)=\frac{3}{2}\unicode[STIX]{x1D6FD}_{\Vert }$ , $(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{s}^{2}(0)=1$ and the correct perpendicular limit $(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{f}^{2}(\unicode[STIX]{x03C0}/2)=1+\unicode[STIX]{x1D6FD}_{\Vert }$ , $(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{s}^{2}(\unicode[STIX]{x03C0}/2)=0$ . If the perpendicular fast mode limit is sufficient to be $(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{f}^{2}(\unicode[STIX]{x03C0}/2)=\unicode[STIX]{x1D6FD}_{\Vert }$ (which is acceptable since $\unicode[STIX]{x1D6FD}_{\Vert }\gg 1$ ), only the terms proportional to $\unicode[STIX]{x1D6FD}_{\Vert }$ can be retained and the fast mode dispersion relation can be simplified to

(3.151) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\Vert }\gg 1:\quad (\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{f}^{2}={\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FD}_{\Vert }\left[1+{\textstyle \frac{1}{2}}\cos ^{2}\unicode[STIX]{x1D703}+\sqrt{\left(1-{\textstyle \frac{5}{2}}\cos ^{2}\unicode[STIX]{x1D703}\right)^{2}+\sin ^{2}\unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D703}}\right].\end{eqnarray}$$

The expression (3.140) yields the velocity ratio of the fast mode as

(3.152) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\Vert }\gg 1:\quad \left(\frac{u_{x}}{u_{z}}\right)^{f}=\frac{1-\frac{5}{2}\cos ^{2}\unicode[STIX]{x1D703}+\sqrt{\left(1-\frac{5}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)^{2}+\sin ^{2}\unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D703}}}{\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}},\end{eqnarray}$$

that is independent of $\unicode[STIX]{x1D6FD}_{\Vert }$ . For the slow mode wave the dispersion relation cannot be further simplified because if we collect only terms proportional to $\unicode[STIX]{x1D6FD}_{\Vert }$ , the resulting expression will be non-zero everywhere except exactly at $\unicode[STIX]{x1D703}=0$ (and also $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ ). For $\unicode[STIX]{x1D703}=0$ , the entire contribution of $(\widetilde{\unicode[STIX]{x1D714}}/\widetilde{k})_{s}^{2}(0)=1$ comes from terms in (3.149) that are not proportional to $\unicode[STIX]{x1D6FD}_{\Vert }$ . Nevertheless, the analytic velocity ratio (3.152) for the fast mode is sufficiently simple and to obtain analytic $\unicode[STIX]{x1D712}_{u}^{s}(\unicode[STIX]{x1D703})$ for the slow mode, it is actually possible to cheat a little by realizing that the slow and fast mode curves are always symmetric and that $\unicode[STIX]{x1D712}_{u}^{s}(\unicode[STIX]{x1D703})+\unicode[STIX]{x1D712}_{u}^{f}(\unicode[STIX]{x1D703})=1$ . Our final large beta results therefore are

(3.153) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}_{u}^{f}(\unicode[STIX]{x1D703})=\left[1+\left(\frac{1-\frac{5}{2}\cos ^{2}\unicode[STIX]{x1D703}+\sqrt{\left(1-\frac{5}{2}\cos ^{2}\unicode[STIX]{x1D703}\right)^{2}+\sin ^{2}\unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D703}}}{\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}}\right)^{2}\right]^{-1}; & \displaystyle\end{eqnarray}$$

(3.154) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}_{u}^{s}(\unicode[STIX]{x1D703})=1-\unicode[STIX]{x1D712}_{u}^{f}(\unicode[STIX]{x1D703}). & \displaystyle\end{eqnarray}$$

Solutions are plotted in figure 3 as dashed lines and fit the solutions of the full dispersion relations with $\unicode[STIX]{x1D6FD}_{\Vert }=10^{3}$ very nicely.

Figure 3. CGL velocity ratio $\unicode[STIX]{x1D712}_{u}(\unicode[STIX]{x1D703})=|u_{\Vert }|^{2}/|u|^{2}$ for the slow mode (red lines) and the fast mode (blue lines) for different values of $\unicode[STIX]{x1D6FD}_{\Vert }$ , with isotropic temperature $T_{\bot }=T_{\Vert }$ (or the temperature anisotropy ratio $a_{p}=1$ ). Panel (a) is for $\unicode[STIX]{x1D6FD}_{\Vert }<2/3$ , and the thickness of the lines increases with $\unicode[STIX]{x1D6FD}_{\Vert }$ when approaching the critical value of $\unicode[STIX]{x1D6FD}_{\Vert }=2/3$ . The plotted lines have $\unicode[STIX]{x1D6FD}_{\Vert }=0.5;0.6;0.64;0.66;0.666$ . Panel (b) is for $\unicode[STIX]{x1D6FD}_{\Vert }>2/3$ and the thickness of the lines decreases with $\unicode[STIX]{x1D6FD}_{\Vert }$ when going away from the critical value of $\unicode[STIX]{x1D6FD}_{\Vert }=2/3$ . The plotted lines have $\unicode[STIX]{x1D6FD}_{\Vert }=0.67;0.7;0.8;1.0;10,10^{3}$ . In both figures, the dotted lines are the critical case $\unicode[STIX]{x1D6FD}_{\Vert }=2/3$ with solutions (3.155). In (b) the dashed lines represent the limit $\unicode[STIX]{x1D6FD}_{\Vert }\gg 1$ with solutions (3.153), (3.154) and the dashed lines nicely overlap with a thin solid line solutions obtained for $\unicode[STIX]{x1D6FD}_{\Vert }=10^{3}$ . The Alfvén mode has $\unicode[STIX]{x1D712}_{u}(\unicode[STIX]{x1D703})=0$ regardless of the plasma beta and is not plotted.

3.10.1 CGL special case $\unicode[STIX]{x1D6FD}_{\Vert }=2/(4-T_{\bot }/T_{\Vert })$

In this peculiar case when all 3 modes propagate in the parallel direction with the same phase speed, one again faces complications when calculating the velocity eigenvector, as with the MHD peculiar case when $V_{A}=C_{s}$ . This CGL special case can be encountered only for $\unicode[STIX]{x1D6FD}_{\Vert }\geqslant 1/2$ because the temperature anisotropy must be $a_{p}\geqslant 0$ . We start with the velocity eigenvector matrix (3.138) and for the special case considered $1+\frac{1}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-2\unicode[STIX]{x1D6FD}_{\Vert }=0$ , the expression (3.139) simplifies to $a-c=(4\unicode[STIX]{x1D6FD}_{\Vert }-1)\sin ^{2}\unicode[STIX]{x1D703}$ . One can already see that for strictly parallel propagation the entire matrix will be zero and it will not be possible to evaluate the eigenvector. Similarly to the MHD case, the $\sin \unicode[STIX]{x1D703}$ can be pulled out of the matrix or alternatively one can use (3.140), (3.141) and calculate the limit. For the special case considered here, the parameter $b=\frac{1}{2}(4\unicode[STIX]{x1D6FD}_{\Vert }-2)\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}$ . Considering again for simplicity only positive wavenumbers, the expression (3.140) yields in this particular case

(3.155) $$\begin{eqnarray}\frac{u_{x}}{u_{z}}=\frac{4\unicode[STIX]{x1D6FD}_{\Vert }-1}{4\unicode[STIX]{x1D6FD}_{\Vert }-2}\tan \unicode[STIX]{x1D703}\pm \sqrt{\left(\frac{4\unicode[STIX]{x1D6FD}_{\Vert }-1}{4\unicode[STIX]{x1D6FD}_{\Vert }-2}\right)^{2}\tan ^{2}\unicode[STIX]{x1D703}+1}.\end{eqnarray}$$

The expression can be evaluated at $\unicode[STIX]{x1D703}=0$ , which yields for the slow mode $(u_{x}/u_{z})^{s}(0)=-1$ and for the fast mode $(u_{x}/u_{z})^{f}(0)=+1$ , implying that both modes start at

(3.156) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{\Vert }=2/(4-a_{p}):\quad \unicode[STIX]{x1D712}_{u}^{s}(0)={\textstyle \frac{1}{2}};\quad \unicode[STIX]{x1D712}_{u}^{f}(0)={\textstyle \frac{1}{2}}.\end{eqnarray}$$

For the perpendicular propagation the discussion is unchanged from the previous general case and yields $\unicode[STIX]{x1D712}_{u}^{s}(\unicode[STIX]{x03C0}/2)=1$ and $\unicode[STIX]{x1D712}_{u}^{f}(\unicode[STIX]{x03C0}/2)=0$ .

3.11 Empirical models with polytropic indices $\unicode[STIX]{x1D6FE}_{\Vert }$ , $\unicode[STIX]{x1D6FE}_{\bot }$

The differences between MHD and CGL can be further clarified with polytropic indices. The linearized MHD pressure equation reads

(3.157) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D6FE}p^{(0)}(\unicode[STIX]{x2202}_{x}u_{x}+\unicode[STIX]{x2202}_{y}u_{y}+\unicode[STIX]{x2202}_{z}u_{z})=0.\end{eqnarray}$$

One can introduce the concept of polytropic indices $\unicode[STIX]{x1D6FE}_{\Vert }$ and $\unicode[STIX]{x1D6FE}_{\bot }$ to the CGL pressure equations (3.73), (3.74) and write

(3.158) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}p_{\Vert }}{\unicode[STIX]{x2202}t}+p_{\Vert }^{(0)}(\unicode[STIX]{x2202}_{x}u_{x}+\unicode[STIX]{x2202}_{y}u_{y})+\unicode[STIX]{x1D6FE}_{\Vert }p_{\Vert }^{(0)}\unicode[STIX]{x2202}_{z}u_{z}=0; & \displaystyle\end{eqnarray}$$

(3.159) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}p_{\bot }}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D6FE}_{\bot }p_{\bot }^{(0)}(\unicode[STIX]{x2202}_{x}u_{x}+\unicode[STIX]{x2202}_{y}u_{y})+p_{\bot }^{(0)}\unicode[STIX]{x2202}_{z}u_{z}=0, & \displaystyle\end{eqnarray}$$

where for the CGL model $\unicode[STIX]{x1D6FE}_{\Vert }=3$ and $\unicode[STIX]{x1D6FE}_{\bot }=2$ .Footnote ⁷ The reasoning for such a construction is to enable a smooth transition between two extreme cases – the CGL model on one side, and the isothermal model $\unicode[STIX]{x1D6FE}_{\Vert }=1$ and $\unicode[STIX]{x1D6FE}_{\bot }=1$ on the other side. The important observation is that in the parallel pressure equation the $\unicode[STIX]{x1D6FE}_{\Vert }$ acts only on the parallel velocity component $u_{z}$ , and in the perpendicular pressure equation the $\unicode[STIX]{x1D6FE}_{\bot }$ acts only on the perpendicular velocity components $u_{x},u_{y}$ . In MHD, the $\unicode[STIX]{x1D6FE}$ acts on all three components $u_{x},u_{y},u_{z}$ . The values of polytropic indices are directly related to the number of degrees of freedom $i$ , through the well-known relation $\unicode[STIX]{x1D6FE}=(i+2)/i$ . In the MHD description the number of degrees of freedom is $i=3$ and so $\unicode[STIX]{x1D6FE}=5/3$ . In contrast, the CGL description can be interpreted as being composed of a (strongly coupled) mixture of one-dimensional and two-dimensional dynamics, where for the parallel direction $i=1$ and $\unicode[STIX]{x1D6FE}_{\Vert }=3$ and for the perpendicular direction $i=2$ and $\unicode[STIX]{x1D6FE}_{\bot }=2$ . The apparently subtle, but nevertheless fundamental difference between (3.157) and (3.158), (3.159) is the core of many differences between the MHD and CGL descriptions, such as for example the effect of the slow mode becoming faster than the Alfvén mode. The two systems remain different even if the mean CGL pressures (or temperatures) are prescribed to be isotropic $p_{\Vert }^{(0)}=p_{\bot }^{(0)}$ and equal to the mean MHD pressure $p^{(0)}$ . This means that the MHD does not match the CGL description even if the distribution function is isotropic, as for example Maxwellian.

We briefly consider models with polytropic indices $\unicode[STIX]{x1D6FE}_{\Vert },\unicode[STIX]{x1D6FE}_{\bot }$ that are based on the linearized pressure equations (3.158), (3.159). If these pressure equations are ‘un-linearized’, i.e. if one performs a ‘reverse engineering’ procedure to obtain nonlinear equations (a procedure that is possible to do because of the known guidance by the CGL model), the nonlinear equations read

(3.160) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}p_{\Vert }}{\text{d}t}+p_{\Vert }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}+(\unicode[STIX]{x1D6FE}_{\Vert }-1)p_{\Vert }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}=0; & \displaystyle\end{eqnarray}$$

(3.161) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}p_{\bot }}{\text{d}t}+\unicode[STIX]{x1D6FE}_{\bot }p_{\bot }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}-(\unicode[STIX]{x1D6FE}_{\bot }-1)p_{\bot }\hat{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{b}}=0, & \displaystyle\end{eqnarray}$$

which can be rewritten as

(3.162) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{p_{\Vert }|\boldsymbol{B}|^{\unicode[STIX]{x1D6FE}_{\Vert }-1}}{\unicode[STIX]{x1D70C}^{\unicode[STIX]{x1D6FE}_{\Vert }}}\right)=0;\quad \frac{\text{d}}{\text{d}t}\left(\frac{p_{\bot }}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|^{\unicode[STIX]{x1D6FE}_{\bot }-1}}\right)=0. & & \displaystyle\end{eqnarray}$$

Empirical fluid models of this type were studied for example by Hau & Sonnerup (Reference Hau and Sonnerup1993) and Hau et al. (Reference Hau, Phan, Sonnerup and Paschmann1993) with the motivation that spacecraft observations in the Earth’s magnetosheath often show behaviour that cannot be explained by the CGL values of $\unicode[STIX]{x1D6FE}_{\Vert }=3,\unicode[STIX]{x1D6FE}_{\bot }=2$ or by the isothermal values of $\unicode[STIX]{x1D6FE}_{\Vert }=1,\unicode[STIX]{x1D6FE}_{\bot }=1$ , but can be fitted by the values that lie somewhere in between (and actually quite close to the isothermal case). For a direct comparison with the CGL model, the equations (3.162) can be rewritten to a form

(3.163) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{p_{\Vert }|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{3}}\right)=-(\unicode[STIX]{x1D6FE}_{\Vert }-3)\left(\frac{p_{\Vert }|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{3}}\right)\frac{\unicode[STIX]{x1D70C}}{|\boldsymbol{B}|}\frac{\text{d}}{\text{d}t}\left(\frac{|\boldsymbol{B}|}{\unicode[STIX]{x1D70C}}\right); & \displaystyle\end{eqnarray}$$

(3.164) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\left(\frac{p_{\bot }}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\right)=(\unicode[STIX]{x1D6FE}_{\bot }-2)\left(\frac{p_{\bot }}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\right)\frac{1}{|\boldsymbol{B}|}\frac{\text{d}}{\text{d}t}|\boldsymbol{B}|, & \displaystyle\end{eqnarray}$$

or to an interesting alternative form

(3.165) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\ln \left(\frac{p_{\Vert }|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{3}}\right)=-(\unicode[STIX]{x1D6FE}_{\Vert }-3)\frac{\text{d}}{\text{d}t}\ln \left(\frac{|\boldsymbol{B}|}{\unicode[STIX]{x1D70C}}\right); & \displaystyle\end{eqnarray}$$

(3.166) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\ln \left(\frac{p_{\bot }}{\unicode[STIX]{x1D70C}|\boldsymbol{B}|}\right)=(\unicode[STIX]{x1D6FE}_{\bot }-2)\frac{\text{d}}{\text{d}t}\ln |\boldsymbol{B}|. & \displaystyle\end{eqnarray}$$

Obviously, for CGL values of $\unicode[STIX]{x1D6FE}_{\Vert }=3,\unicode[STIX]{x1D6FE}_{\bot }=2$ the right-hand sides disappear and one obtains the classical CGL expressions. The right-hand sides of the above expressions can be compared with equations of Chew et al. (Reference Chew, Goldberger and Low1956) equations (2.82), (2.83) which contain the gyrotropic heat flux contributions. Therefore, the deviations from the CGL values of $\unicode[STIX]{x1D6FE}_{\Vert }=3,\unicode[STIX]{x1D6FE}_{\bot }=2$ can be considered to offer a very simple empirical model (perhaps too simple) that shows how the heat flux contributions modify the CGL dynamics. The dispersion relations for this fluid model can be found in Hau et al. (Reference Hau, Phan, Sonnerup and Paschmann1993) and also in Abraham-Shrauner (Reference Abraham-Shrauner1973), who studied an even more general empirical model with four polytropic indices (essentially, the $\unicode[STIX]{x1D6FE}_{\Vert }$ , $\unicode[STIX]{x1D6FE}_{\bot }$ acting on $|\boldsymbol{B}|$ and $\unicode[STIX]{x1D70C}$ in (3.162) can be further split to two independent parameters). Interestingly, complex polytropic indices have been suggested to model Landau damping (Belmont & Mazelle Reference Belmont and Mazelle1992). Considering the linearized CGL equations (3.68)–(3.72) and linearized polytropic pressure equations (3.158)–(3.159) written in the $x$ – $z$ plane yields the dispersion relations for this model. The Alfvén wave is still decoupled from the rest of the system and the Alfvén wave dispersion relation is unaffected by the different pressure equations, therefore also yielding the same oblique firehose instability threshold. What is modified is the dispersion relation of the slow and fast waves. The velocity equations read

(3.167) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}^{2}u_{x}}{\unicode[STIX]{x2202}t^{2}}-\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}(\unicode[STIX]{x1D6FE}_{\bot }\unicode[STIX]{x2202}_{x}^{2}u_{x}+\unicode[STIX]{x2202}_{x}\unicode[STIX]{x2202}_{z}u_{z})+\left(\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}-\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}-V_{A}^{2}\right)\unicode[STIX]{x2202}_{z}^{2}u_{x}-V_{A}^{2}\unicode[STIX]{x2202}_{x}^{2}u_{x}=0; & \displaystyle\end{eqnarray}$$

(3.168) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}^{2}u_{z}}{\unicode[STIX]{x2202}t^{2}}-\unicode[STIX]{x1D6FE}_{\Vert }\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}\unicode[STIX]{x2202}_{z}^{2}u_{z}-\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}\unicode[STIX]{x2202}_{x}\unicode[STIX]{x2202}_{z}u_{x}=0, & \displaystyle\end{eqnarray}$$

and the transformation to Fourier space yields the velocity matrix

(3.169) $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}\displaystyle \unicode[STIX]{x1D714}^{2}-\left(\unicode[STIX]{x1D6FE}_{\bot }\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}+V_{A}^{2}\right)k_{\bot }^{2}+\left(\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}-\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}-V_{A}^{2}\right)k_{\Vert }^{2}; & \displaystyle -\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}k_{\bot }k_{\Vert }\\ \displaystyle -\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}k_{\bot }k_{\Vert }; & \displaystyle \unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D6FE}_{\Vert }\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}k_{\Vert }^{2}\end{array}\right)\left(\begin{array}{@{}c@{}}u_{x}\\ u_{z}\end{array}\right)=\left(\begin{array}{@{}c@{}}0\\ 0\end{array}\right), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

or alternatively

(3.170) $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}\displaystyle \widetilde{\unicode[STIX]{x1D714}}^{2}-\left(\unicode[STIX]{x1D6FE}_{\bot }a_{p}\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}+1\right)\widetilde{k}_{\bot }^{2}-\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)\right)\widetilde{k}_{\Vert }^{2}; & \displaystyle -\frac{1}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\widetilde{k}_{\bot }\widetilde{k}_{\Vert }\\ \displaystyle -\frac{1}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\widetilde{k}_{\bot }\widetilde{k}_{\Vert }; & \displaystyle \widetilde{\unicode[STIX]{x1D714}}^{2}-\unicode[STIX]{x1D6FE}_{\Vert }\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}\widetilde{k}_{\Vert }^{2}\end{array}\right)\left(\begin{array}{@{}c@{}}u_{x}\\ u_{z}\end{array}\right)=\left(\begin{array}{@{}c@{}}0\\ 0\end{array}\right). & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Calculating the determinant yields a dispersion relation for the slow and fast modes in the form

(3.171) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D714}}^{4}-A_{2}\widetilde{\unicode[STIX]{x1D714}}^{2}+A_{0}=0; & \displaystyle\end{eqnarray}$$

(3.172) $$\begin{eqnarray}\displaystyle & \displaystyle A_{2}=\widetilde{k}_{\bot }^{2}\left(1+\frac{\unicode[STIX]{x1D6FE}_{\bot }}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\right)+\widetilde{k}_{\Vert }^{2}\left(1+\frac{1}{2}\unicode[STIX]{x1D6FD}_{\Vert }(\unicode[STIX]{x1D6FE}_{\Vert }-1)+\frac{1}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\right); & \displaystyle\end{eqnarray}$$

(3.173) $$\begin{eqnarray}\displaystyle & \displaystyle A_{0}=\frac{\unicode[STIX]{x1D6FE}_{\Vert }}{2}\widetilde{k}_{\Vert }^{2}\unicode[STIX]{x1D6FD}_{\Vert }\left[\widetilde{k}_{\Vert }^{2}\left(1-\frac{1}{2}\unicode[STIX]{x1D6FD}_{\Vert }+\frac{1}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\right)+\widetilde{k}_{\bot }^{2}\left(1+\frac{\unicode[STIX]{x1D6FE}_{\bot }}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-\frac{1}{2\unicode[STIX]{x1D6FE}_{\Vert }}a_{p}^{2}\unicode[STIX]{x1D6FD}_{\Vert }\right)\right]. & \displaystyle\end{eqnarray}$$

Solutions for parallel propagation $(k_{\bot }=0)$ are the usual CGL Alfvén mode and the ion-acoustic mode $\widetilde{\unicode[STIX]{x1D714}}=\pm \sqrt{(\unicode[STIX]{x1D6FE}_{\Vert }/2)\unicode[STIX]{x1D6FD}_{\Vert }}\widetilde{k}_{\Vert }$ , and for perpendicular propagation $(k_{\Vert }=0)$ the fast mode $\widetilde{\unicode[STIX]{x1D714}}=\pm \sqrt{1+(\unicode[STIX]{x1D6FE}_{\bot }/2)a_{p}\unicode[STIX]{x1D6FD}_{\Vert }}\widetilde{k}_{\bot }$ . Similarly to the CGL description it can be shown that $A_{2}^{2}-4A_{0}\geqslant 0$ because of the trick (3.137), implying that the slow mode becomes unstable when $A_{0}<0$ , i.e. when

(3.174) $$\begin{eqnarray}\left[k_{\Vert }^{2}\left(1-\frac{1}{2}\unicode[STIX]{x1D6FD}_{\Vert }+\frac{1}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }\right)+k_{\bot }^{2}\left(1+\frac{\unicode[STIX]{x1D6FE}_{\bot }}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-\frac{1}{2\unicode[STIX]{x1D6FE}_{\Vert }}a_{p}^{2}\unicode[STIX]{x1D6FD}_{\Vert }\right)\right]<0.\end{eqnarray}$$

The two competing instabilities are again the parallel firehose instability and the highly oblique mirror instability. The parallel firehose instability has the same threshold as obtained previously. However, the mirror instability threshold reads

(3.175) $$\begin{eqnarray}1+\frac{\unicode[STIX]{x1D6FE}_{\bot }}{2}a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-\frac{1}{2\unicode[STIX]{x1D6FE}_{\Vert }}a_{p}^{2}\unicode[STIX]{x1D6FD}_{\Vert }<0,\end{eqnarray}$$

which is consistent with the result of Hau & Sonnerup (Reference Hau and Sonnerup1993), their equation (14), written in the formFootnote ⁸

(3.176) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{\Vert }\unicode[STIX]{x1D6FD}_{\Vert }<\frac{\unicode[STIX]{x1D6FD}_{\bot }^{2}}{2+\unicode[STIX]{x1D6FE}_{\bot }\unicode[STIX]{x1D6FD}_{\bot }}.\end{eqnarray}$$

For $\unicode[STIX]{x1D6FE}_{\Vert }=3,\unicode[STIX]{x1D6FE}_{\bot }=2$ , the condition (3.175) is naturally equivalent to the CGL mirror threshold. However, the condition offers a temptingly simple result to keep the perpendicular polytropic index unchanged $\unicode[STIX]{x1D6FE}_{\bot }=2$ and modifies the parallel polytropic index to $\unicode[STIX]{x1D6FE}_{\Vert }=1/2$ . In this way, the mirror threshold matches the correct kinetic threshold (3.94). Nevertheless, this is just an interesting empirical concept, it fixes the highly oblique mirror threshold but at the same time moves the slow and fast mode dispersion relations in other directions, for example the parallel sound speed in such a model is $C_{\Vert }^{2}=\frac{1}{2}p_{\Vert }^{(0)}/\unicode[STIX]{x1D70C}_{0}$ , which appears to be unrealistically low. Such a ‘tuning’ of models with free parameters – only in one particular direction and only at the level of the dispersion relations – can be sometimes useful for interpretation of observational data, however, such models will usually lead to unphysical situations. We will not consider models with polytropic indices further in this text. Instead, to obtain a better match with kinetic description, it is advisable to use the correct CGL values of $\unicode[STIX]{x1D6FE}_{\Vert }=3,\unicode[STIX]{x1D6FE}_{\bot }=2$ and focus on fluid models that derive the correct heat flux contributions $q_{\Vert }$ and $q_{\bot }$ .

3.12 CGL and radially expanding solar wind

Similarly to MHD, the CGL fluid model can be very useful for understanding the evolution of temperature in the radially expanding solar wind. A good discussion can be found for example in Matteini et al. (Reference Matteini, Landi, Hellinger, Pantellini, Maksimovic, Velli, Goldstein and Marsch2007), Matteini et al. (Reference Matteini, Hellinger, Goldstein, Landi, Velli and Neugebauer2013) and references therein. Let us first consider the MHD case. Combining the pressure equation $\text{d}p/\text{d}t+\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0$ with the density equation $\text{d}\unicode[STIX]{x1D70C}/\text{d}t+\unicode[STIX]{x1D70C}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0$ yields

(3.177) $$\begin{eqnarray}\text{MHD:}\quad \frac{\text{d}}{\text{d}t}\left(\frac{p}{\unicode[STIX]{x1D70C}^{\unicode[STIX]{x1D6FE}}}\right)=0;\quad \Rightarrow \quad \frac{\text{d}}{\text{d}t}\left(\frac{T}{\unicode[STIX]{x1D70C}^{\unicode[STIX]{x1D6FE}-1}}\right)=0,\end{eqnarray}$$

which in a steady state (or co-moving with the convective derivative) implies $T\sim \unicode[STIX]{x1D70C}^{\unicode[STIX]{x1D6FE}-1}$ . By considering simple radially expanding solar wind, one prescribes that the density $\unicode[STIX]{x1D70C}$ evolves with the heliocentric distance $r$ as $\unicode[STIX]{x1D70C}\sim 1/r^{2}$ , and by using $\unicode[STIX]{x1D6FE}=5/3$ , the ideal MHD model yields (in the absence of any dissipation and associated turbulent heating)

(3.178) $$\begin{eqnarray}T\sim r^{-4/3}.\end{eqnarray}$$

In contrast, the CGL equations (3.41) are rewritten as

(3.179) $$\begin{eqnarray}\text{CGL:}\quad \frac{\text{d}}{\text{d}t}\left(\frac{T_{\Vert }|\boldsymbol{B}|^{2}}{\unicode[STIX]{x1D70C}^{2}}\right)=0;\quad \frac{\text{d}}{\text{d}t}\left(\frac{T_{\bot }}{|\boldsymbol{B}|}\right)=0,\end{eqnarray}$$

which in a steady state implies

(3.180) $$\begin{eqnarray}T_{\Vert }\sim \frac{\unicode[STIX]{x1D70C}^{2}}{|\boldsymbol{B}|^{2}};\quad T_{\bot }\sim |\boldsymbol{B}|;\quad \frac{T_{\bot }}{T_{\Vert }}=\frac{|\boldsymbol{B}|^{3}}{\unicode[STIX]{x1D70C}^{2}}.\end{eqnarray}$$

For the magnetic field it is possible to consider profiles of roughly $|\boldsymbol{B}|\sim 1/r^{2}$ , or $|\boldsymbol{B}|\sim 1/r$ (one can be more precise and consider the Parker spiral profile). The first magnetic field profile yields

(3.181) $$\begin{eqnarray}|\boldsymbol{B}|\sim 1/r^{2};\quad \Rightarrow \quad T_{\Vert }\sim \text{const.};\quad T_{\bot }\sim 1/r^{2};\quad \frac{T_{\bot }}{T_{\Vert }}\sim 1/r^{2}.\end{eqnarray}$$

The second magnetic field profile yields

(3.182) $$\begin{eqnarray}|\boldsymbol{B}|\sim 1/r;\quad \Rightarrow \quad T_{\Vert }\sim 1/r^{2};\quad T_{\bot }\sim 1/r;\quad \frac{T_{\bot }}{T_{\Vert }}\sim r.\end{eqnarray}$$

There is also a curious case when the magnetic field profile is prescribed to be $|\boldsymbol{B}|\sim r^{-4/3}$ , i.e. a profile between two cases of $1/r$ and $1/r^{2}$ , that yields

(3.183) $$\begin{eqnarray}|\boldsymbol{B}|\sim r^{-4/3};\quad \Rightarrow \quad T_{\Vert }\sim r^{-4/3};\quad T_{\bot }\sim r^{-4/3};\quad \frac{T_{\bot }}{T_{\Vert }}\sim \text{const.},\end{eqnarray}$$

and the temperature anisotropy stays constant. The estimations are of course valid only until a firehose threshold or a mirror threshold is reached. For example, evolution according to (3.181) will lead the system to a firehose threshold, and evolution according to (3.182) will lead the system to a mirror threshold. In general, if we fix the density profile to $\unicode[STIX]{x1D70C}\sim 1/r^{2}$ , any magnetic field profile steeper than $r^{-4/3}$ will evolve the system towards a firehose threshold, and a magnetic field profile shallower than $r^{-4/3}$ will evolve the system towards a mirror threshold. The estimations do not reveal the presence of the firehose instability or the mirror instability in the CGL model, since the solutions are written in a steady state, and concern only evolution of mean temperatures (i.e. the solutions should be written with $T_{\Vert }^{(0)},T_{\bot }^{(0)}$ ). To find an instability, one of course needs to consider evolution equations for fluctuating variables and to analyse the associated dispersion relations, since it is the fluctuations that become unstable. Further discussion can be found in Hunana & Zank (Reference Hunana and Zank2017).

From a linear perspective, the CGL model has to be accompanied by the Hall term and FLR corrections, so that the instabilities are stabilized at small spatial scales. However, considering nonlinear evolution with finite amplitudes of fluctuations, even the non-dispersive CGL model is stabilized. For example, considering parallel propagation, Tenerani, Velli & Hellinger (Reference Tenerani, Velli and Hellinger2017) has shown that the firehose instability criterion reads (see (9) in that paper)

(3.184) $$\begin{eqnarray}V_{A}^{2}+\frac{1}{\unicode[STIX]{x1D70C}_{0}}\frac{p_{\bot }^{(0)}-p_{\Vert }^{(0)}}{1+\left|\displaystyle \frac{\boldsymbol{B}_{\bot }}{B_{0}}\right|^{2}}<0;\quad \Rightarrow \quad 1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}\frac{a_{p}-1}{1+\left|\displaystyle \frac{\boldsymbol{B}_{\bot }}{B_{0}}\right|^{2}}<0,\end{eqnarray}$$

i.e. for infinitely small amplitudes of the magnetic field, the usual firehose criterion is obtained. However, when the CGL system reaches firehose-unstable regime and the amplitude of the fluctuations starts to grow, the firehose criterion is modified according to (3.184). Therefore, the amplitude of the fluctuations cannot grow without bounds, and beyond some critical amplitude, the system becomes again stabilized. The condition (3.184) nicely demonstrates how the CGL system is stabilized during nonlinear evolution, once the firehose instability is reached. For further discussion, see Tenerani et al. (Reference Tenerani, Velli and Hellinger2017), Tenerani & Velli (Reference Tenerani and Velli2018).

To finish this subsection, the heuristically generalized CGL model with polytropic indices $\unicode[STIX]{x1D6FE}_{\Vert }$ , $\unicode[STIX]{x1D6FE}_{\bot }$ yields that, in the steady state, the mean temperatures evolve according to

(3.185) $$\begin{eqnarray}T_{\Vert }=\left(\frac{\unicode[STIX]{x1D70C}}{|\boldsymbol{B}|}\right)^{\unicode[STIX]{x1D6FE}_{\Vert }-1};\quad T_{\bot }=|\boldsymbol{B}|^{\unicode[STIX]{x1D6FE}_{\bot }-1};\quad \frac{T_{\bot }}{T_{\Vert }}=\frac{|\boldsymbol{B}|^{\unicode[STIX]{x1D6FE}_{\Vert }+\unicode[STIX]{x1D6FE}_{\bot }-2}}{\unicode[STIX]{x1D70C}^{\unicode[STIX]{x1D6FE}_{\Vert }-1}},\end{eqnarray}$$

and such heuristic models can yield a wide array of possible temperature profiles.

3.13 CGL model – summary

The usual ‘combining’ of evolution equations (by performing $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t$ and substituting one equation into another) is very useful to gain insight into the linear eigenmodes of a considered system. However, for more advanced fluid models, such a procedure can become very analytically involved, and it is much easier to calculate the determinant of the entire system with analytic software, such as Maple or Mathematica. Therefore, for more advanced fluid models we often do not write down the final dispersion relation for arbitrary propagation angle (which might be uneconomically large to write down), and we only provide normalized, linearized in the $x$ – $z$ plane and Fourier transformed equations. For easy comparison with more advanced fluid models, it is beneficial to write the final CGL equations in the same form. It is useful to work in normalized units, and we drop writing the tilde. The normalized, linearized in the $x$ – $z$ plane, and Fourier transformed CGL equations read

(3.186) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D714}\unicode[STIX]{x1D70C}+k_{\bot }u_{x}+k_{\Vert }u_{z}=0; & \displaystyle\end{eqnarray}$$

(3.187) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D714}u_{x}+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}k_{\bot }p_{\bot }-v_{A\Vert }^{2}k_{\Vert }B_{x}+k_{\bot }B_{z}=0; & \displaystyle\end{eqnarray}$$

(3.188) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D714}u_{y}-v_{A\Vert }^{2}k_{\Vert }B_{y}=0; & \displaystyle\end{eqnarray}$$

(3.189) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D714}u_{z}+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}k_{\Vert }p_{\Vert }+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(1-a_{p})k_{\bot }B_{x}=0; & \displaystyle\end{eqnarray}$$

(3.190) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D714}B_{x}-k_{\Vert }u_{x}=0;\quad -\unicode[STIX]{x1D714}B_{y}-k_{\Vert }u_{y}=0;\quad -\unicode[STIX]{x1D714}B_{z}+k_{\bot }u_{x}=0; & \displaystyle\end{eqnarray}$$

(3.191) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D714}p_{\Vert }+k_{\bot }u_{x}+3k_{\Vert }u_{z}=0; & \displaystyle\end{eqnarray}$$

(3.192) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D714}p_{\bot }+2a_{p}k_{\bot }u_{x}+a_{p}k_{\Vert }u_{z}=0, & \displaystyle\end{eqnarray}$$

where we have introduced the normalized parallel Alfvén speed  $v_{A\Vert }^{2}\equiv 1+(\unicode[STIX]{x1D6FD}_{\Vert }/2)(a_{p}-1)$ (the tilde on $v_{A\Vert }$ is dropped too). The determinant can be easily calculated and factorized in Maple, yielding the CGL dispersion relation in normalized units (tilde are dropped)

(3.193) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D714}^{2}-k_{\Vert }^{2}v_{A\Vert }^{2})(\unicode[STIX]{x1D714}^{4}-A_{2}\unicode[STIX]{x1D714}^{2}+A_{0})=0; & \displaystyle\end{eqnarray}$$

(3.194) $$\begin{eqnarray}\displaystyle & \displaystyle A_{2}=k_{\bot }^{2}(1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert })+k_{\Vert }^{2}\left(v_{A\Vert }^{2}+{\textstyle \frac{3}{2}}\unicode[STIX]{x1D6FD}_{\Vert }\right); & \displaystyle\end{eqnarray}$$

(3.195) $$\begin{eqnarray}\displaystyle & \displaystyle A_{0}={\textstyle \frac{3}{2}}k_{\Vert }^{2}\unicode[STIX]{x1D6FD}_{\Vert }\left[k_{\Vert }^{2}v_{A\Vert }^{2}+k_{\bot }^{2}\left(1+a_{p}\unicode[STIX]{x1D6FD}_{\Vert }-{\textstyle \frac{1}{6}}a_{p}^{2}\unicode[STIX]{x1D6FD}_{\Vert }\right)\right], & \displaystyle\end{eqnarray}$$

which of course agrees with the previously obtained CGL solutions for the Alfvén mode (3.81), and the slow and fast modes (3.90).

4.1 Hall-CGL model with cold electrons

Here, we want to study the simplest Hall-CGL model for the proton species. We neglect the electron inertia and we make the electrons cold with $\boldsymbol{p}_{e}=0$ . The full model is described by the usual CGL equations (3.36)–(3.39), but the induction equation now reads

(4.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times (\boldsymbol{u}_{p}\times \boldsymbol{B})-\frac{c}{4\unicode[STIX]{x03C0}e}\unicode[STIX]{x1D735}\times \left(\frac{1}{n}(\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}\right).\end{eqnarray}$$

The induction equation is linearized as

(4.11) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}\overset{\text{lin}}{=}\unicode[STIX]{x1D735}\times (\boldsymbol{u}_{p}\times \boldsymbol{B}_{0})-\frac{c}{4\unicode[STIX]{x03C0}en_{0}}\unicode[STIX]{x1D735}\times ((\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}_{0}).\end{eqnarray}$$

By using (3.67), direct calculation of the second term yields

(4.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\times ((\unicode[STIX]{x1D735}\times \boldsymbol{B})\times \boldsymbol{B}_{0})=B_{0}\left(\begin{array}{@{}c@{}}\unicode[STIX]{x2202}_{y}\unicode[STIX]{x2202}_{z}B_{z}-\unicode[STIX]{x2202}_{z}^{2}B_{y}\\ -\unicode[STIX]{x2202}_{x}\unicode[STIX]{x2202}_{z}B_{z}+\unicode[STIX]{x2202}_{z}^{2}B_{x}\\ \unicode[STIX]{x2202}_{x}\unicode[STIX]{x2202}_{z}B_{y}-\unicode[STIX]{x2202}_{y}\unicode[STIX]{x2202}_{z}B_{x}\end{array}\right), & & \displaystyle\end{eqnarray}$$

and by using $c/(4\unicode[STIX]{x03C0}en_{0})=V_{A}^{2}/(\unicode[STIX]{x1D6FA}_{p}B_{0})$ , the Hall-term contributions are

(4.13) $$\begin{eqnarray}\displaystyle c\unicode[STIX]{x1D735}\times \boldsymbol{E}_{H}\overset{\text{lin}}{=}\frac{V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}}\left(\begin{array}{@{}c@{}}\unicode[STIX]{x2202}_{y}\unicode[STIX]{x2202}_{z}B_{z}-\unicode[STIX]{x2202}_{z}^{2}B_{y}\\ -\unicode[STIX]{x2202}_{x}\unicode[STIX]{x2202}_{z}B_{z}+\unicode[STIX]{x2202}_{z}^{2}B_{x}\\ \unicode[STIX]{x2202}_{x}\unicode[STIX]{x2202}_{z}B_{y}-\unicode[STIX]{x2202}_{y}\unicode[STIX]{x2202}_{z}B_{x}\end{array}\right). & & \displaystyle\end{eqnarray}$$

The entire linearized induction equation reads

(4.14) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}\overset{\text{lin}}{=}B_{0}\left(\begin{array}{@{}c@{}}\unicode[STIX]{x2202}_{z}u_{x}\\ \unicode[STIX]{x2202}_{z}u_{y}\\ -\unicode[STIX]{x2202}_{x}u_{x}-\unicode[STIX]{x2202}_{y}u_{y}\end{array}\right)-\frac{V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}}\left(\begin{array}{@{}c@{}}\unicode[STIX]{x2202}_{y}\unicode[STIX]{x2202}_{z}B_{z}-\unicode[STIX]{x2202}_{z}^{2}B_{y}\\ -\unicode[STIX]{x2202}_{x}\unicode[STIX]{x2202}_{z}B_{z}+\unicode[STIX]{x2202}_{z}^{2}B_{x}\\ \unicode[STIX]{x2202}_{x}\unicode[STIX]{x2202}_{z}B_{y}-\unicode[STIX]{x2202}_{y}\unicode[STIX]{x2202}_{z}B_{x}\end{array}\right), & & \displaystyle\end{eqnarray}$$

which when written in the $x$ – $z$ plane (all $\unicode[STIX]{x2202}_{y}=0$ ) yields

(4.15) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}B_{x}}{\unicode[STIX]{x2202}t}=B_{0}\unicode[STIX]{x2202}_{z}u_{x}+\frac{V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}}\unicode[STIX]{x2202}_{z}^{2}B_{y}; & \displaystyle\end{eqnarray}$$

(4.16) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}B_{y}}{\unicode[STIX]{x2202}t}=B_{0}\unicode[STIX]{x2202}_{z}u_{y}+\frac{V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}}(\unicode[STIX]{x2202}_{x}\unicode[STIX]{x2202}_{z}B_{z}-\unicode[STIX]{x2202}_{z}^{2}B_{x}); & \displaystyle\end{eqnarray}$$

(4.17) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}B_{z}}{\unicode[STIX]{x2202}t}=-B_{0}\unicode[STIX]{x2202}_{x}u_{x}-\frac{V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}}\unicode[STIX]{x2202}_{x}\unicode[STIX]{x2202}_{z}B_{y}. & \displaystyle\end{eqnarray}$$

These equations replace the $\unicode[STIX]{x2202}B/\unicode[STIX]{x2202}t$ equations in the linearized CGL system written in the $x$ – $z$ plane. And the trouble is immediately apparent. In contrast to the CGL system, we are no longer able to separate the Alfvén mode which was before exclusively in the $B_{y}$ , $u_{y}$ components. Now, all the components are coupled and all three modes, the Alfvén mode, the slow mode and the fast mode are coupled. This is not surprising, since the same situation is in the Hall-MHD. In this case, we have no other choice and we have to calculate the dispersion relation for all three modes. The dispersion relation can be solved analytically only for special cases and, in general, the dispersion relation has to be solved numerically. Let us quickly consider the two special cases of parallel and perpendicular propagation that can be solved analytically.

4.2 Parallel propagation

For propagation parallel to the mean magnetic field, all $\unicode[STIX]{x2202}_{x}=0$ and the familiar sound mode/ion-acoustic mode decouples in the $u_{z}$ , $p_{\Vert }$ components as

(4.18) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}u_{z}}{\unicode[STIX]{x2202}t^{2}}-\frac{3p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}\unicode[STIX]{x2202}_{z}^{2}u_{z}=0,\end{eqnarray}$$

with the familiar dispersion relation $\unicode[STIX]{x1D714}^{2}=C_{\Vert }^{2}k_{\Vert }^{2}$ . The dispersion relation of this mode does not depend on the Hall term. However, the other two modes are now coupled through

(4.19) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}u_{x}}{\unicode[STIX]{x2202}t}-\frac{1}{B_{0}}\left(V_{A}^{2}-\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}+\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}\right)\unicode[STIX]{x2202}_{z}B_{x}=0; & \displaystyle\end{eqnarray}$$

(4.20) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}u_{y}}{\unicode[STIX]{x2202}t}-\frac{1}{B_{0}}\left(V_{A}^{2}-\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}+\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}\right)\unicode[STIX]{x2202}_{z}B_{y}=0; & \displaystyle\end{eqnarray}$$

(4.21) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}B_{x}}{\unicode[STIX]{x2202}t}=B_{0}\unicode[STIX]{x2202}_{z}u_{x}+\frac{V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}}\unicode[STIX]{x2202}_{z}^{2}B_{y}; & \displaystyle\end{eqnarray}$$

(4.22) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}B_{y}}{\unicode[STIX]{x2202}t}=B_{0}\unicode[STIX]{x2202}_{z}u_{y}-\frac{V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}}\unicode[STIX]{x2202}_{z}^{2}B_{x}. & \displaystyle\end{eqnarray}$$

If the last terms in the $B$ -field equations are neglected, i.e. if the Hall term is neglected, the two modes decouple and one naturally recovers the CGL parallel propagating Alfvén modes, both propagating with the phase speed

(4.23) $$\begin{eqnarray}v_{A\Vert }^{2}=V_{A}^{2}-\frac{p_{\Vert }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}+\frac{p_{\bot }^{(0)}}{\unicode[STIX]{x1D70C}_{0}}=V_{A}^{2}\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)\right).\end{eqnarray}$$

Applying $\unicode[STIX]{x2202}_{t}$ to the $B$ -field equations yields

(4.24) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}^{2}B_{x}}{\unicode[STIX]{x2202}t^{2}}=v_{A\Vert }^{2}\unicode[STIX]{x2202}_{z}^{2}B_{x}+\frac{V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}}\unicode[STIX]{x2202}_{z}^{2}\frac{\unicode[STIX]{x2202}B_{y}}{\unicode[STIX]{x2202}t}; & \displaystyle\end{eqnarray}$$

(4.25) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}^{2}B_{y}}{\unicode[STIX]{x2202}t^{2}}=v_{A\Vert }^{2}\unicode[STIX]{x2202}_{z}^{2}B_{y}-\frac{V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}}\unicode[STIX]{x2202}_{z}^{2}\frac{\unicode[STIX]{x2202}B_{x}}{\unicode[STIX]{x2202}t}, & \displaystyle\end{eqnarray}$$

which when transformed to Fourier space read

(4.26) $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D714}^{2}-v_{A\Vert }^{2}k_{\Vert }^{2}; & \displaystyle +\text{i}\frac{V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}}k_{\Vert }^{2}\unicode[STIX]{x1D714}\\ \displaystyle -\text{i}\frac{V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}}k_{\Vert }^{2}\unicode[STIX]{x1D714}; & \unicode[STIX]{x1D714}^{2}-v_{A\Vert }^{2}k_{\Vert }^{2}\end{array}\right)\left(\begin{array}{@{}c@{}}B_{x}\\ B_{y}\end{array}\right)=\left(\begin{array}{@{}c@{}}0\\ 0\end{array}\right), & & \displaystyle\end{eqnarray}$$

and the zero determinant requirement implies

(4.27) $$\begin{eqnarray}\unicode[STIX]{x1D714}^{4}-\unicode[STIX]{x1D714}^{2}k_{\Vert }^{2}\left(2v_{A\Vert }^{2}+k_{\Vert }^{2}\frac{V_{A}^{4}}{\unicode[STIX]{x1D6FA}_{p}^{2}}\right)+k_{\Vert }^{4}v_{A\Vert }^{4}=0.\end{eqnarray}$$

The solutions of the dispersion relation (quadratic polynomial in $\unicode[STIX]{x1D714}^{2}$ ) is simply

(4.28) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{W}^{2}=k_{\Vert }^{2}V_{A}^{2}\left[1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)+\frac{k_{\Vert }^{2}V_{A}^{2}}{2\unicode[STIX]{x1D6FA}_{p}^{2}}+\frac{|k_{\Vert }|V_{A}}{\unicode[STIX]{x1D6FA}_{p}}\sqrt{1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)+\frac{k_{\Vert }^{2}V_{A}^{2}}{4\unicode[STIX]{x1D6FA}_{p}^{2}}}\right]; & & \displaystyle\end{eqnarray}$$

(4.29) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{IC}^{2}=k_{\Vert }^{2}V_{A}^{2}\left[1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)+\frac{k_{\Vert }^{2}V_{A}^{2}}{2\unicode[STIX]{x1D6FA}_{p}^{2}}-\frac{|k_{\Vert }|V_{A}}{\unicode[STIX]{x1D6FA}_{p}}\sqrt{1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)+\frac{k_{\Vert }^{2}V_{A}^{2}}{4\unicode[STIX]{x1D6FA}_{p}^{2}}}\right], & & \displaystyle\end{eqnarray}$$

where, importantly, $\sqrt{k_{\Vert }^{2}}=|k_{\Vert }|$ was used. The solution (4.28) is the whistler mode, and the solution (4.29) is the ion-cyclotron mode. For isotropic temperatures ( $a_{p}=1$ ), the result is equal to the Hall-MHD dispersion relation. At long wavelengths ( $k_{\Vert }\rightarrow 0$ ) the solutions become non-dispersive and connect to the usual CGL Alfvén modes. For parallel propagation, the Hall term is therefore responsible for splitting the two modes that otherwise would propagate at the same speed, while it does not influence the ion-acoustic mode. Or in another words, using the vocabulary of slow, Alfvén, fast, depending on plasma $\unicode[STIX]{x1D6FD}_{\Vert }$ and temperature anisotropy $a_{p}$ , the Hall term either splits the parallel propagating Alfvén and fast mode, or the slow and Alfvén mode. At very short spatial scales $(k_{\Vert }\gg 1)$ , the limits areFootnote ⁹

(4.30) $$\begin{eqnarray}k_{\Vert }\rightarrow \pm \infty :\quad \unicode[STIX]{x1D714}_{IC}^{2}=\unicode[STIX]{x1D6FA}_{p}^{2}\left(1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)\right)^{2};\quad \unicode[STIX]{x1D714}_{W}^{2}=k_{\Vert }^{4}\frac{V_{A}^{4}}{\unicode[STIX]{x1D6FA}_{p}^{2}}.\end{eqnarray}$$

For isotropic temperatures $a_{p}=1$ the frequency of the ion-cyclotron mode (that should be actually called the proton-cyclotron mode) converges towards the proton-cyclotron frequency $\pm \unicode[STIX]{x1D6FA}_{p}$ . For anisotropic temperatures, the frequency of the mode starts to be $\unicode[STIX]{x1D6FD}_{\Vert }$ dependent and will converge to frequencies that can be higher or lower than $\unicode[STIX]{x1D6FA}_{p}$ . The frequency of the whistler mode is not bounded and technically goes to infinity, which is a consequence of neglecting the electron inertia. Importantly, both modes are stable for high wavenumbers. The solutions (4.28), (4.29) are analytically correct, but similarly to the Hall-MHD, the solutions can be rewritten to a more useful form, by realizing that the dispersion equation (4.27) can be decomposed asFootnote ¹⁰

(4.31) $$\begin{eqnarray}\left(\unicode[STIX]{x1D714}^{2}+\frac{k_{\Vert }^{2}V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}}\unicode[STIX]{x1D714}-k_{\Vert }^{2}v_{A\Vert }^{2}\right)\left(\unicode[STIX]{x1D714}^{2}-\frac{k_{\Vert }^{2}V_{A}^{2}}{\unicode[STIX]{x1D6FA}_{p}}\unicode[STIX]{x1D714}-k_{\Vert }^{2}v_{A\Vert }^{2}\right)=0.\end{eqnarray}$$

The first bracket yields two solutions

(4.32) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{IC,1} & = & \displaystyle -\frac{k_{\Vert }^{2}V_{A}^{2}}{2\unicode[STIX]{x1D6FA}_{p}}+|k_{\Vert }|V_{A}\sqrt{1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)+\frac{k_{\Vert }^{2}V_{A}^{2}}{4\unicode[STIX]{x1D6FA}_{p}^{2}}};\quad -\text{Green line};\end{eqnarray}$$

(4.33) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{W,1} & = & \displaystyle -\frac{k_{\Vert }^{2}V_{A}^{2}}{2\unicode[STIX]{x1D6FA}_{p}}-|k_{\Vert }|V_{A}\sqrt{1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)+\frac{k_{\Vert }^{2}V_{A}^{2}}{4\unicode[STIX]{x1D6FA}_{p}^{2}}};\quad -\text{dashed Magenta line},\end{eqnarray}$$

and the second bracket yields two solutions

(4.34) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{W,2} & = & \displaystyle +\frac{k_{\Vert }^{2}V_{A}^{2}}{2\unicode[STIX]{x1D6FA}_{p}}+|k_{\Vert }|V_{A}\sqrt{1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)+\frac{k_{\Vert }^{2}V_{A}^{2}}{4\unicode[STIX]{x1D6FA}_{p}^{2}}};\quad -\text{Blue line};\end{eqnarray}$$

(4.35) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{IC,2} & = & \displaystyle +\frac{k_{\Vert }^{2}V_{A}^{2}}{2\unicode[STIX]{x1D6FA}_{p}}-|k_{\Vert }|V_{A}\sqrt{1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)+\frac{k_{\Vert }^{2}V_{A}^{2}}{4\unicode[STIX]{x1D6FA}_{p}^{2}}};\quad -\text{dashed Cyan line}.\end{eqnarray}$$

It is straightforward to check that $(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{W,1})(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{W,2})=0$ yields the whistler mode solution (4.28), and $(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{IC,1})(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{IC,2})=0$ yields the ion-cyclotron mode solution (4.29). Often, solutions (4.32)–(4.35) are written in an alternative form where the absolute value on $k_{\Vert }$ is removed (which obviously can be done), and the solutions are just slightly re-arranged. Additionally, for more complicated fluid models, we want to write solution in one line instead of four lines. In this example, we just write a shortcut that two solutions are

(4.36) $$\begin{eqnarray}\unicode[STIX]{x1D714}=\pm \frac{k_{\Vert }^{2}V_{A}^{2}}{2\unicode[STIX]{x1D6FA}_{p}}+k_{\Vert }V_{A}\sqrt{1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)+\frac{k_{\Vert }^{2}V_{A}^{2}}{4\unicode[STIX]{x1D6FA}_{p}^{2}}},\end{eqnarray}$$

with another two solutions obtained by substituting $\unicode[STIX]{x1D714}$ with $-\unicode[STIX]{x1D714}$ .

4.2.1 Firehose instability

Now we can discuss the firehose instability. If the quantity $1+(\unicode[STIX]{x1D6FD}_{\Vert }/2)(a_{p}-1)>0$ , then all the solutions (4.32)–(4.35) have frequencies that are purely real for the entire range of wavenumbers. For isotropic temperatures $a_{p}=1$ , the dispersion relations do not depend on the value of $\unicode[STIX]{x1D6FD}_{\Vert }$ , which is a consequence of neglecting the FLR pressure corrections, as will be discussed later, and all solutions are equivalent to the Hall-MHD dispersion relations. Solutions (4.32)–(4.35) with $a_{p}=1$ are plotted in figure 4(a), where we plotted both positive and negative wavenumbers and positive and negative frequencies to clearly understand how the modes are connected to each other. The magenta and cyan lines are plotted as dashed lines, so that the growth rates in the firehose-unstable regime (plotted in figure 5) are clearly visible. For anisotropic temperatures $a_{p}\neq 1$ , the solutions are naturally $\unicode[STIX]{x1D6FD}_{\Vert }$ dependent. To show solutions in a firehose unstable regime, one has to choose $\unicode[STIX]{x1D6FD}_{\Vert }>2$ since the instability does not exist for lower values of $\unicode[STIX]{x1D6FD}_{\Vert }$ . In figure 4(b), we have chosen a value of $\unicode[STIX]{x1D6FD}_{\Vert }=4$ and varied the temperature anisotropy from $a_{p}=2$ down to a critical value of $a_{p}=0.5$ . In this regime, all the modes have purely real frequencies.

Figure 4. Dispersion relations for the Hall-CGL model (with cold electrons), for parallel propagation $\unicode[STIX]{x1D703}=0$ . The colour coding of lines is according to (4.32)–(4.35). (a) Isotropic temperatures $a_{p}=T_{\bot }/T_{\Vert }=1.0$ . The dispersion relations do not depend on the value of $\unicode[STIX]{x1D6FD}_{\Vert }$ , which is a consequence of neglecting the FLR pressure corrections, etc. The figure is actually equivalent to the Hall-MHD model. (b) the value of $\unicode[STIX]{x1D6FD}_{\Vert }$ is fixed with the value $\unicode[STIX]{x1D6FD}_{\Vert }=4.0$ and the temperature anisotropy is varied as $a_{p}=2.0;1.0;0.5$ . The thickness of the curves increases with decreasing $a_{p}$ as we approach the firehose threshold at $a_{p}=0.5$ . All the curves have purely real frequency $\unicode[STIX]{x1D714}$ .

Importantly, if the temperature anisotropy parameter $a_{p}$ is further decreased, the frequencies of all the modes will become complex numbers (with a real and imaginary part) in the region where

(4.37) $$\begin{eqnarray}1+\frac{\unicode[STIX]{x1D6FD}_{\Vert }}{2}(a_{p}-1)+\frac{k_{\Vert }^{2}V_{A}^{2}}{4\unicode[STIX]{x1D6FA}_{p}^{2}}<0,\end{eqnarray}$$

which can be viewed as a modified Hall-CGL firehose instability threshold that is now length-scale dependent. At long spatial scales ( $k_{\Vert }\rightarrow 0$ ), the above criterion is naturally equivalent to the CGL firehose criterion. The term proportional to $k_{\Vert }^{2}$ in (4.37) is always positive and as $k_{\Vert }$ increases, after some critical wavenumber the frequency of modes will become purely real. The Hall term is therefore responsible for the stabilization of the firehose instability at sufficiently high wavenumbers (small spatial scales). The solutions in this firehose-unstable regime are plotted in figure 5, where again $\unicode[STIX]{x1D6FD}_{\Vert }=4$ and the temperature anisotropy is varied from $a_{p}=0.49$ to the lowest possible value of $a_{p}=0.0$ . Figure 5(a) shows real frequencies and the (b) shows imaginary frequencies.

Figure 5. Dispersion relations for the Hall-CGL model (with cold electrons), for parallel propagation $\unicode[STIX]{x1D703}=0$ , with fixed $\unicode[STIX]{x1D6FD}_{\Vert }=4$ , after crossing the firehose threshold $a_{p}=T_{\bot }/T_{\Vert }<0.5$ . The colour coding of lines is according to (4.32)–(4.35). The thickness of the lines decreases as going away from the threshold $a_{p}=0.5$ . (a) Real frequency and the curves have $a_{p}=0.49;0.2;0.0$ . (b) Imaginary frequency and the curves have $a_{p}=0.4;0.2;0.0$ . The mode with $a_{p}=0.49$ , shown in (a), has an imaginary part of the frequency which is very close to zero for the entire range of $k_{\Vert }$ . For this reason this mode is not plotted in (b), where instead a mode with $a_{p}=0.4$ is plotted. A mode that has $\text{Im}\,\unicode[STIX]{x1D714}>0$ is unstable and growing in time. A mode that has $\text{Im}\,\unicode[STIX]{x1D714}<0$ is stable and damped. According to the figure, whistler modes with $\unicode[STIX]{x1D714}_{r}>0$ are unstable, and ion-cyclotron modes with $\unicode[STIX]{x1D714}_{r}<0$ are unstable. It is assumed that $\sqrt{-1}=+\text{i}$ . The solutions for $\unicode[STIX]{x1D714}_{r}<0$ are here non-causal, and (b) should be re-plotted with causal solutions (4.43)–(4.46), so that whistler modes are always unstable, and ion-cyclotron modes stable.

It is of interest to determine which mode becomes unstable. According to our Fourier transform (see appendix A), a mode with positive imaginary frequency is unstable and growing, and a mode with negative imaginary frequency is stable and damped. If the firehose threshold (4.37) is satisfied, for some range of $k_{\Vert }$ the second terms in solutions (4.32)–(4.35) become purely imaginary and the first terms stay purely real. Consider one of the four possible quadrants in figure 5(a), for example the quadrant with $k_{\Vert }>0;\unicode[STIX]{x1D714}_{r}>0$ . Up to some critical $k_{\Vert }$ where the Hall-CGL firehose criterion is satisfied, both the blue mode and the cyan mode propagate with the same real frequency $\unicode[STIX]{x1D714}_{r}=+k_{\Vert }^{2}V_{A}^{2}/(2\unicode[STIX]{x1D6FA}_{p})$ . The differences are in the imaginary frequency, see figure 5(b), where in this quadrant the blue mode is unstable and the cyan mode is stable (and damped). After a critical $k_{\Vert }$ (for example for $a_{p}=0$ it is $k_{\Vert }V_{A}/\unicode[STIX]{x1D6FA}_{p}=2$ ), both modes become purely real and evolve with different real frequencies, here the blue mode is obviously the whistler mode and the cyan mode is the ion-cyclotron mode. The same situation is in the quadrant $k_{\Vert }<0;\unicode[STIX]{x1D714}_{r}>0$ . We can therefore conclude that for positive real frequencies, $\unicode[STIX]{x1D714}_{r}>0$ , the whistler mode is unstable and the ion-cyclotron mode is stable and damped.

Now we could say that considering both positive and negative wavenumbers and frequencies is redundant, and that we are finished. However, from a perspective of this guide, we consider useful to show that one might easily come into contradictions, if negative frequencies are not handled with sufficient care. Importantly, according to figure 5, for negative real frequencies $\unicode[STIX]{x1D714}_{r}<0$ , the whistler mode is stable and the ion-cyclotron mode is unstable. This is indeed contradictory. Keeping the $k_{\Vert }>0$ , a change from positive to negative $\unicode[STIX]{x1D714}_{r}$ represents a change of direction of propagation along $B_{0}$ . One can argue, that in a homogeneous system (e.g. excluding propagation in a stratified fluid), a change of direction of propagation should not yield a change of stability. Additionally, kinetic theory and simulations show, that it is the whistler mode that is firehose unstable for both $\unicode[STIX]{x1D714}_{r}>0$ and $\unicode[STIX]{x1D714}_{r}<0$ , and the ion-cyclotron mode is stable. It is important to emphasize that, from a perspective of solutions (4.32)–(4.35), at the range of wavenumbers where the firehose instability is present, the whistler and ion-cyclotron modes propagate (in a given quadrant) with the same real frequency. Moreover, both modes have the same polarization of electric and magnetic field (see § 4.2.3). Both modes are completely degenerate, and not distinguishable. Thus, how the modes are continued for higher wavenumbers once the firehose instability disappears, is not obvious (since $\sqrt{-1}$ can be possibly both $+\text{i}$ or $-\text{i}$ ). One should consider all the possible solutions and verify, which solutions are physically plausible. Therefore, after the firehose threshold is crossed, the analytic solutions (4.32)–(4.35) are not correctly separated. Obviously, more ‘physical information’ is needed to distinguish between the ion-cyclotron and whistler modes in the firehose-unstable regime, and to determine which solutions are causal and which are non-causal.